So this is looking in on America and maybe this isn't the way it really is but it seems like the two big cultural fault lines they have are guns and abortion. And it appears that people are either pro gun and anti abortion or anti gun and pro abortion.
How did those two end up linked in this way? My assumption would be if you were "Pro-Life" you would be against guns, and if you were "Pro-Choice" you would be for guns?
It’s largely a consequence of partisan politics and the development of the Republican party in the latter half of the twentieth century. The pro-gun part of the party is an outgrowth of the joining together of the John Birch (extremist anti-Communist) wing of the Republican party with the NRA to promote an ahistorical unlimited right to bear arms (see for example). In the 1970s, in the wake of Roe v Wade, the Republican party took this as an opportunity to try to peel off support for the Democratic party from Catholics who had previously been a reliable part of the Democratic coalition. At the same time, with Nixon’s Southern strategy which aimed to move Southern whites disgruntled by the civil rights advances of the 1960s meant that many protestant denominations, who had previously viewed abortion as a “Catholic issue” and were either neutral or pro-abortion adopted an anti-abortion viewpoint.
Not too surprisingly, the root cause of both of these was a reaction towards the civil rights bills and other advances for Black civil rights of the 1960s. Part of how protestants were inveigled into an anti-abortion stance was the promise that reversing Roe v. Wade could also lead to reversing the decisions that led to Bob Jones v. United States.
In simple terms
I agree with wbg's answer. I would add that, in general, Republicans tend to think of the Constitution as a static document and Democrats tend to view the Constitution as fluid. These interpretations, mixed with religious tendencies (informing the definition of "life") lead to the clustering of issues you're describing.
Broader Context
The two major political parties in the United States have different Overton Windows, or "regions of acceptable discourse."
Using the corporate/legacy media, here are some examples:
If a person who supports abortion appears on CNN, MSNBC, a late-night talk show, etc. and voices support for guns, they will be ostracized, laughed at, booed, etc. This is not allowed in the mainstream Left's Overton Window.
Likewise, if a pro-life person on Fox News, etc. voices support for the anti-gun movement, they will be ostracized, booed, etc. This position is not allowed in the mainstream Right's Overton Window.
Overton windows are largely shaped by Media* and culture, which flows downstream into politics. People who do not fit neatly into this categories can feel "politically homeless" and forced to vote for the lesser of two evils, or not at all.
Because corporate media requires this predictable orthodoxy, finding dissenting voices often requires other mediums such as podcasts, online videos, blogs etc. In these domains, there is a much larger Overton Window.
*You may find the idea of the Press-Controlled State intriguing, especially in this context.
Summary
There is an ideological basis for how a given group manifests its political positions. However, the positions that are presented through media, are not as organic as they appear. CGCampbell's comment eludes to this:
It doesn't. I am (very) pro-gun and (very) pro-choice. Stop lumping all people into slots.
In addition to the answers presented which focus on political strategy, I present some background on the lesser understood portions of the conservative position which when added to the "main stream" definition represents a large segment of the US.
Many conservatives in America believe in a very small central government and look to the US Constitution for guidance on all matters of law and even culture. They tend to not accept modern categories of "human rights", such as a right to health care or housing for instance. Only the core rights are considered as necessary for a free society.
One connection between the two issues, is that the unborn are considered citizens and thus protected by the Constitution. Likewise the right to keep and bear arms is directly encoded.
The recently overturned SCOTUS decisions allows states to create laws that govern essentially, when the fetus is a protected citizen and when it is not. It is considered un-constitutional for the federal government to make a blanket rule on behalf of the citizens of states. The citizens must vote in their state to encode the when the fetus is considered a citizen and when it is not.
In short, the common thread is the constitutional legality of the two issues.
Don's answer about partisan politics starts us off in the right direction. But the red herring in the question is that these two issues are not linked in any special way.
In a culture of partisan politics, each issue gets divided into camps that are motivated to split far apart to carve out as large a chunk for themselves as they can get, as discussed in this question. This pull means that issues will find themselves with strange bedfellows. Consider this list:
Each of these issues, and many more, has become strongly polarized. But it's not the issues' fault. It's the climate in which everything must belong to one camp or the other. Hence, even these factors get drawn in:
(See this spreadsheet collating some exit poll data from the 2020 election, which brings up at least one other interesting corollary issue: voting based on platform vs. voting based on candidate...)
The last few years it's been interesting to see that even some things that were considered nonpartisan are becoming so (e.g. Russia, race relations, police funding).
And while you could probably draw a causal connection between some of these issues, such as an anti-abortion stance being tied to a Christian belief in the existence of the soul from conception, you find the opposite relationship for other issues, such as degree of interest in socioeconomic equality. In short, the politics outweighs the causal / natural relationship between issues (though the latter might gently tug a new issue in one direction or the other).
And so you get "pro guns" and "anti abortion" on one camp, and you get Twitter commentators arguing that the term "pro life" can't apply to both, and there's sense to that because they really aren't united by one coherent ideology.
Note two caveats: (1) Subjectively, I find that most people are not as polarized in their day-to-day lives and relationships as they are in their political leaning, much as the sports team you cheer for rarely affects your friendships. (2) Parties carving up issues to attract voters does not necessarily mean enacting policy. When the camps have such strong loyalty and are so pitted against one another, it takes a lot to lose a follower. And the actual laws that get passed are much closer to centre than the avowed positions. (That's also because of the semi-functional checks and balances, of course.)
These two caveats help explain why the Roe v. Wade decision is so surprising and why even Trump realizes that actually making such moves is politically risky...
The big question I have following this analysis is how the US got so polarized in the first place. This is dealt with a little in the question linked above. Many authors also link an apparent acceleration in the last couple of decades with the rise of social media, from the bubble effect to the algorithmic preference for anger and fear as drivers of engagement with ads. All beyond my pay grade, though.
Though most answers are already very good to the question, I feel I can maybe add something.
For now, let's ignore the two points presented in the question and instead focus on the roots of these opinions. That is, understand why someone becomes pro-life/pro-choice and pro-gun/anti-gun.
Many people, mainly conservatives, believe that the constitution should be respected above all else, not necessarily just because it is the constitution, but because they believe in the moral principles the constitution was written on top of. As there is an article exclusively stating that the right to bear arms should always be protected, people that respect the constitution tend to be heavily pro-gun.
People that are anti-gun tend to be more of a pragmatic type of person, and generally follow the line of thought that if only good and trained people had the right to carry firearms, it would result in less incidents involving them.
About the abortion topic, there is still a lot of discussion. Many people who believe in the moral principles of natural rights (life, liberty and property), at least in my perspective, are religious (catholics) and thus don't support abortion due to religion. That not the only point pro-life people have though.
One that was already mentioned here is that once a female egg has been fertilized by a male sperm, that cell is essencially a human being, and therefore has the natural right to live. That is also in the US constitution, by the way.
But there are pro-guns that are also pro-choice, being the less common of the "stereotypes". One of their arguments, is that even though the baby has the right to live, the mother also has the right of not wanting another individual inside her and living off of their nutrients, and shall be removed.
Abortion is still a highly debated topic philosophically and people who believe in natural rights still haven't developed a definitive answer for this question, as it implies a lot of things that may contradict themselves.
The US being essencially a system of two parties, everyone must concentrate on just one of the two sides of the conflict of opinions, even though they may not fully agree with the side they're on. This even makes people be in contact more frequently with more extreme opinions of the side they're on, further increasing polarization, which is why you probably asked the question you posted here.
NOTE: I'm saying this from my own research on natural rights, phylosophy, and my own perspective of things, being one of the pro-gun and pro-abortion ones.
Just so it's said, one of the core features of nationalist movements is masculinism: the lionization of men and masculine virtues held as a core tenet. This is part of the psychological dynamic of nationalism. Nationalist leaders create a sense of insecurity through xenophobic messaging and visions of lost power and heritage, then use that insecurity to bind followers to the nationalist cause. Masculinism is part of that second move, where nationalists call up 'real' men who can help recover what was stolen from them. It's an effective (if manipulative) strategy.
Masculinism entails both an affection for ostentatious displays of power and authority and a belief in a subordination and disempowerment of women. Pro-gun and anti-abortion attitudes are typical of the insecure male desire to attain, control, and defend what is 'his by right' but has been (to the nationalist mindset) denied to him.
The right-nationalist swing in US conservatism over the last 30 years or so means that conservatives have come to hammer on these particular issues. Not because the issues themselves are popular or essential — both issues have minority support in the US — but because these issues are critical for maintaining the insecurity-driven masculinism that's central to the nationalist programme.
