Is quantile regression a maximum likelihood method? Quantile regression allows to estimate a conditional quantile for y (like e.g. the median of y,...) from data x.
I do not see any distributional assumptions about y being made. This seems in contrast to maximum likelihood estimation which starts with making an assumption about the distribution of y (e.g gaussian distribution).
Therefore the question: is quantile regression a maximum likelihood method?
If not, what is the broader term for methods like quantile regression?

Additional rewording:
What is the rationale of the quantile loss function in quantile regression (of arbitrary complex models)? Does it rely on the specification of the distributional form of response variable? And, specifically, is the quantile loss (somehow) a (log) likelihood function?

It depends on the loss function you're trying to minimize. In MLE you're not always assuming that the distribution is gaussian. For instance, there's a relation between the distribution you assume and the loss function you try to minimize

*

*Gaussian distribution $\propto e^{-(x-\mu)^2}$ implies $L^2$ loss

*Laplace distribution $\propto e^{-|x-\mu|}$ implies $L^1$ loss

In the case of Gaussian distribution
$$
P(y|x) = N(y| f(x;\theta), \sigma^2)
$$
where $\sigma$ is fixed. Therefore the likelihood estimation is is
$$
\theta^* = \text{arg max}_\theta \prod_i P(y_i|x_i) = \text{arg max}_\theta \sum_i \log P(x_i | y_i)
$$
and therefore
$$
\theta^* = \text{arg max} -n \log \sigma - \frac{n}{2} \log (2\pi) -\frac{1}{2} \sum_i \left(\frac{y_i - f(x_i, \theta)}{\sigma}\right)^2
$$
which removing constant terms is
$$
\theta^* = \text{argmax}_\theta -\sum_i \left(y_i - f(x_i, \theta)\right)^2
$$
which is the standard $L^2$ loss function for regression problems. So, as you can see by defining a loss function you're also defining which distribution you assume of your data.
In the case of quantile regression is the same. Depending on your loss function you'll implicitly be specifying an underlying distribution.
