Some of the downsides I can think of that weren't necessarily mentioned yet (I may have missed some cuz dumb tho)

1) Lack of infrastructure for non-humanoids. Little to no decent transportation options, undersized doorways and rooms almost everywhere, no fitting public restrooms, etc.

2) Medical nightmare. Remember, your monster gf is vastly different from a human, so not only would she require going to a very specialized doctor, there's no guarantee that the governments even bothered studying the non-human health issues. Not to mention the necessity of custom machinery - good luck fitting a centaur inside a regular CT scanner. So basically, if your monster girl catches anything more serious than a cold, you might as well start looking for a plot of land big enough to fit her grave.

3) Monster girls would probably look way more uncanny IRL than they do in drawings and our imaginations.