Square works just as well.

Not really. Using a parallelogram means each x-coordinate’s allowed range depends on the y-coordinate and vice versa, but using a square, the coordinates are independent.

Affine transformations don't change relative density.

You can think of it this way. There's a density function on the shapes in question. Whenever you transform the 2d space, you have to adjust the density at the same time to preserve "volume" (area times density).

Non-linear transforms, such as interpreting a square as polar coordinates to obtain a disk, will expand or shrink area differently in different parts of the space, which means that if you start with a uniform density, you end up with a non-uniform density. But linear/affine transforms affect area the same everywhere in the space, and so if the density is uniform to begin with, it remains uniform.

I believe an affine transformation would keep density perfectly consistent across the whole area.

no, this person is not suggesting a parallelogram but rather performing an affine transformation on the triangle to make it a right triangle so they can pick a random point in a square instead. as another commenter mentioned, I believe this distorts the distribution and won't do the right thing.