It's probably hard to give a global answer to a Q like this, but from
Anti-Feminism: four strategies for the demonisation and depoliticisation of feminism on Chinese social media [...]
four strategies used to demonise feminists and depoliticise feminism online in China are identified: feminists as deviant women, as betraying the nation, as connected to Islamists, and as “fake-feminists.” The article highlights a kind of intertwined anti-feminism that draws power from distinct features—nationalism and Islamophobia.
The most common form of online abuse directed at women feminists involved criticising their appearance, marital status and personality. For example, the following comment was made on an influential grassroots feminist account:
Judging from the women I personally know, no women who call themselves feminists have good looks, have a good personality or receive love from men. Feminists are pathetic. They are all ugly and do not enjoy any gender privileges. Feminism is good because it shows who female losers are. (February 28 2016, 03:19)
[...]
Chinese anti-feminism and online misogyny are closely entwined with nationalist discourse, which is a site for accomplishing hegemonic masculinity (Joane Nagel 1998). This connection arguably legitimises anti-feminist rhetoric and attacks on women as a patriotic defence of, and sign of commitment to, the nation. For example, an influential feminist account posted a Virginia Woolf quote on Woolf’s birthday: “As a woman, I have no country. As a woman, I want no country. As a woman, my country is the whole world” (January 25 2018, 14:57). Some anti-feminists used this post as evidence to argue that acceptance of feminism constitutes betrayal of the nation:
We can see that feminism has been completely developed in the direction of anti-nation, anti-state, and anti-Chinese men. When feminism, which should fight for the legitimate rights and interests of women of our country, has completely deviated from the bottom line of our nation, then this feminist power has become the greatest evil. (August 11 2018, 22:33)
[...]
feminists are easily slandered as haters of Chinese men and worshippers of Western men. A good example of this is when a feminist account published a post encouraging more tolerance of diversity in intimate relationships (such as less discrimination against divorced women and inter-racial marriage) in a Western context, and criticised Chinese men for their intolerance of Chinese women marrying foreigners.This post attracted many anti-feminist comments:
This is a group of fucking bitches who worship foreigners. Get out of China … Our great motherland is not where you traitors should stay. Go lick your foreign sugar-daddy, idiot. (February 3 2019, 19:02)
[...]
anti-feminists’ nationalist rhetoric is not just based on the pretext of “Western-rooted” characteristics, but is also easily rationalised by the official discourse and actions of the ACWF, which is framed in governmental propaganda as the only correct and suitable organisation to empower women in China. The term “feminism” (and related terms, such as “feminist” and “patriarchy”) has rarely been used in governmental discourse. One of my interviewees, Yvonne, pointed this out when talking about why feminism is highly stigmatised in China:
Feminism in China is easily regarded as a Western anti-China force. However, the term “women’s liberation” [funü jiefang] also comes from the West, from Marxism. Unlike feminism, women’s liberation has been “officially qualified.” (Interview, March 14 2018)
So, yeah, Ted's answer is somewhat correct on this angle. Some (Chinese) nationalists see feminism as a threat to their idea of a nation. But it is slightly more complicated than that:
The phrase “women’s liberation” (funü jiefang) has been used as an official term in governmental discourse which is distinct from feminism (nüquan zhuyi) in the Chinese language. This distinction has an historical legacy in China as feminism was deemed a product of the bourgeoisie, and separated from the socialist women’s liberation (Dongchao Min 2005). Based on socialist ideals, the ACWF tend to carefully to distinguish “Marxist women’s liberation theory” from “Western bourgeois feminism.” [...] In contrast, the ideas circulated through—and the strategies applied by—grassroots feminist activists on social media seemly more closing align with the “Western” way of doing feminism; for example, advocating for “speaking out” in the context of the #MeToo movement, calling for LGBT rights and for policy change, and protesting against sexism in employment and education. These feminist ideas and activities are criticised by opponents as “middle-class oriented;”
More interestingly, perhaps:
The third main strategy in which anti-feminists stigmatise feminism is by linking feminists with Islam and using Islamophobia to stir up public panic about feminists and feminist organisations. Few academic papers have specifically focused on this online phenomenon, but several news reports have covered the linking (disseminated by others) of feminism with Islam. These reports, and the feminist account operators I interviewed, agree that an event in 2014 was the starting point for this specific kind of cyber-attack on feminists and feminist accounts. At a college in Guangzhou, a female Muslim student was asked to take off her headscarf during military training. After she posted about this incident online, some feminists supported her right to choose her own religious beliefs and how to dress. That night, an influential feminist account’s home page was hacked, and a cartoon image on the page was altered so it appeared to be wearing a black burqa. A simplified but “persuasive” rationale was then circulated online: removing the hijab was the liberation of Islamic women while putting on the hijab was a surrender to Islamic culture. [...] After this incident, posts claiming that feminism had colluded with Islam began to circulate on social-media platforms.
[...] The aforementioned three strategies show how anti-feminists or misogynists are aligned with other groups—such as nationalists, Han supremacists and Islamophobes—to form a loose and polycentric alliance.
IMHO most of those broil down to some form of social-dominance orientation. (FWTW, there's even a paper claiming in its title "Social dominance orientation among women is associated with the endorsement of benevolent sexism".)
Anyhow, the more Western perspective is also briefly summarized there:
Banet-Weiser (2018) summarises the logic behind anti-feminism and misogyny as the notion that society has been destroyed by feminists, and men have been “injured by women”.
For the latter, there's a lot more that's been written e.g.
The politics of enmity is central to contemporary conservative and populist movements,
which seek a malign foe against which to posit the promise of national restoration. As
Sanders and Jenkins discuss, right-wing populist ‘retrotopian’ fantasies promise to make
the nation great again while pitting the interests of the ‘pure people’ against a cast of
corrupting enemies, who must be cleansed. Many threatening figures have filled this role
through time: immigrants, racial and religious minorities, socialists, ‘globalists’, the
media, and that most amorphous of categories, ‘elites’.
While populism is not inherently misogynistic (Moghadam and Kaftan 2019; Mudde
and Kaltwasser 2015), today’s right-wing ‘patriarchal populists’ blend populism with
sexism and increasingly deem feminists and sexual minorities dangerous, corrupting
‘enemies of the people’ (Graff and Korolczuk 2021; Kaul 2021; Korolczuk and Graff 2018;
Sanders and Jenkins, this issue). Feminists are accused of emasculating men, lowering
national birth rates, promoting sexual deviance and advancing ‘elite’ interests. False
allegations about the dangers of contraception, abortion or LGBTIQ+ rights generate
moral panics; [...]
Contemporary anti-feminism has a complex posture towards human rights. Political
commitments to hierarchy and biological essentialism push against notions of individual
equality and freedom. Women’s rights are threatening because they disrupt hierarchical
social order and traditional values. [...]
Yet the current contestation and rejection of women’s rights rarely abandons the
concept of rights altogether. Instead, anti-feminists undermine women’s rights by downplaying the legitimacy of women’s rights, by stripping women’s rights of substantive
commitments to gender justice or by invoking competing rights narratives (Bob 2019;
Corredor 2021; Sanders 2018; Schneiker 2019). [...] The Trump
administration’s Commission on Unalienable Rights suggested that women’s and health
rights constituted a new form of ‘ad hoc’ rights meriting less consideration than truly
‘unalienable’rights articulated in the 1776 Bill of Rights or the 1948 Universal Declaration
of Human Rights (see Sanders and Jenkins in this issue). Anti-feminists’ hierarchical
world-view applies not only to societies but also to a hierarchy of rights themselves.
Or if you prefer another source, less abstractly detailing some US viewpoints:
The use of the term “feminism” in this chapter reflects how conservative women activists conceptualize it. [...] Ann Coulter has published numerous books castigating liberals and commented that feminists are “marauding, bloodthirsty vipers” (2005: 325). Another commentator whose career is more closely defined by her ardent attacks on feminists is Christina Hoff Sommers. In Who Stole Feminism? she contends that women have been victimized by “gender” feminists—self-interested, elite, privileged actors who pit women against men (1995). She further chastises feminists’ alleged preoccupation with pain and oppression and argues that most women are not represented by feminists, either within the academy or in national organizations.
That article contends that the message is not so new in US politics, e.g.
opponents blamed feminists for trying to make women’s lives more difficult (Klatch 1987), framed the ERA [Equal Rights Amendment] as a burden not a right, and argued that the ERA “would wipe out the most basic and precious legal right homemakers now enjoy: the right to be a full-time homemaker” (quoted in Marshall 1985: 356). Compounding these explicit anti-feminist arguments, Schlafly and others claimed that feminists were miserable, whiny, and disgruntled (Marshall 1985) and did not represent women. In doing so, the countermovement she spearheaded constructed anti-feminist women as those who could truly speak to women’s interests.
I don't feel like belaboring the point on conscription, since that's been covered at length in another answer, but that situation is not unique to Russia, e.g.
According to a survey conducted in 2019, 34.5% of young [South] Korean
women (aged 19-34) support feminism, compared to 38.7% of young Korean
men who are opposed to it (Ma et al. 2020a, p. 317). Both support for
feminism and hostility towards it are particularly strong among those who
belong to the young generation. Korea’s press and politicians have termed
this phenomenon the “gender conflict (jendeo galdeung) of the young
generation.” However, at the center of men’s hostility towards feminism there
is a public sentiment that the conscription system, which applies only to men
in Korea, is unfair. This impression is directly reflected in lyrics such as “Why
don’t you go to the military?” from a song entitled “Feminist” that was
released in 2018 by a well-known 33-year-old Korean male rapper.
(That paper also has a poll [p. 487] showing that over half of South Korean males think "women should also join the military", with the proportion decreasing by age, from about 2 in 3 in their 20s agreeing with that, to roughly half of the men in their 50s.)