I would remove that vent cover, tile to the outside of the open box that's behind it, and then find a nicer grate cover to go over that.

There are some standard looking ones thrown into this search result, but this should give you some better ideas: Vent Covers

You could take it and have it chrome plated. Look for shops that cater to custom motorcycle folks. You could also strip off the many coats of paint and neatly re-paint it the same color as your tile.

There are places where you can get specialty register covers in various finishes, so you can probably find one to match this shape and whatever color your tile will be. You may need to have the duct end adjusted to handle the additional depth and still give a point for the screws to attach, or it might just work out if the screws have a long bite already. It might be expensive, however, as this looks like an outdated size. Search around a bit.

@acm thanks for the advice on measuring it and finding a replacement. The apartment complex was built in the 70s, so I'm hoping I can find a replacement that's chrome plated because then it'll match my bathroom. I'm going to go test it out tomorrow to see if it's still a functioning vent. If it's not I'll probably just put a cover over it

Is there any other source of air to your bathroom? Or could it be a return air supply for the floor? I'd be very hesitant to cover it up...