You seem to confuse two closely-related yet very different concepts: a regression model (which is a specification of a statistical model) and a parameter estimation method (which essentially is a data-based objective function formulation and subsequent numerical procedures).
For simplicity, we restrict ourselves to the linear parametric family. A quantile regression (model) models (i.e., approximates) the $\tau$-th conditional quantile of the response $y$ given predictors $x$ as a linear function of parameters:
\begin{align}
Q_\tau(y|x) = \alpha + \beta'x. \tag{1}
\end{align}
Likewise, a mean regression (model) models the conditional mean of the response $y$ given predictors $x$ as a linear function of parameters:
\begin{align}
E(y|x) = \alpha + \beta'x. \tag{2}
\end{align}
In principle (of course, it is somewhat a quite narrow view), $(1)$ standalone is the heart of "quantile regression" -- we do not need to understand how parameters $\alpha$ and $\beta$ will be estimated to specify a quantile regression model.
The parameter estimation problem kicks in when a sample $\{(y_i, x_i): i = 1, \ldots, n\}$ is observed. As you may have already known, typically $\alpha$ and $\beta$ are estimated by minimizing the sum of check function:
\begin{align}
(\hat{\alpha}, \hat{\beta}) = \operatorname{argmin}_{\alpha, \beta}\sum_{i = 1}^n \rho_\tau(y_i - \alpha - \beta'x_i), \tag{3}
\end{align}
which is numerically implemented by the linear programming or interior point algorithm. Of course, this is one of many parameter estimation methods (when $\tau = 0.5$, this is usually referred to as Least Absolute Deviation Estimation), which, as you stated, does not require any distributional assumption of $y$.
The MLE, as another parameter estimation method, on the other hand, can be carried out only if one specifies the complete conditional distribution of $y$. That means, to do maximum likelihood estimation, you will need to specify a statistical model that is more granular than regression models such as $(1)$ or $(2)$ (it is more granular because the complete distribution function contains much more information than the quantile function or the mean function only. In fact, both $Q_\tau(y|x)$ and $E(y|x)$ can be derived probabilistically from the distribution of $y|x$). For example, a statistical model like
\begin{align}
y | x \sim f(\alpha + \beta'x; \theta), \tag{4}
\end{align}
where $f$ is some known density function with additional parameter $\theta$. Model $(4)$ then entails the likelihood function (assuming the observations are i.i.d.)
\begin{align}
L(\alpha, \beta; \theta) = \prod_{i = 1}^nf(\alpha + \beta x_i; \theta),
\end{align}
which can be maximized over the parameter space to determine MLE.
It is well-known that when $f$ is Gaussian, Model $(4)$ implies the mean regression model $(2)$, and when $f$ is (asymmetrical) Laplacian, Model $(4)$ implies the quantile regression model $(1)$. For other conditional distributions $f$, in general $(1)$ or $(2)$ are not nested in $(4)$ (that is, neither $Q_\tau(y|x)$ nor $E(y|x)$ admits the simple linear form $\alpha + \beta'x$ under $(4)$).
In summary, the question "Is quantile regression a maximum likelihood method?" is somewhat ill-posed because the former is a statistical model while the latter is a parameter estimation method that depends on a more granular statistical model than the quantile regression model. In this sense, I do not think these two concepts are comparable. If by "what is the broader term for methods like quantile regression?", you meant "what is the parameter estimation method typically used to estimate $(\alpha, \beta)$ in $(1)$?", then the answer is $(3)$ -- I am not sure if there is a universally accepted term for this minimization problem, but it may be OK to call it "least $L^1$ estimation", in view of $\rho_\tau(t) = t(\tau - I_{(-\infty, 0)}(t))$ is a piecewise linear function.
