Comparing these two is like saying a woman wanting to wear a man's shirt is like someone wanting their arm to be invisible.
You're not changing anything about the anatomy of your character by having access to certain clothing or hairstyles. Unless it's any of Minfilia's outfits (which entirely turned me off buying them but that's another topic on its own).
I'm not against making tails invisible, I just don't think it's similar to wanting access to more clothing and hairstyles. That said I am confused why someone would choose a race that has a prominent physical feature they dislike.