Baths and showers create steam and damp air which I would have thought isnt the best condition for fixed carpets.
The floor is the last place to be affected by steam. Hot air rises, and condensation will collect on the ceiling and work it's way down the walls from the top. Ventilation needs to get rid of damp air regardless of whether or not you have carpets.
When I lived alone in a 1989 new-build, I found carpets installed by the developer worked fine in the bathroom. When I left ten years later, their condition was still like new.
I inherited carpets in my next flat which were a bit decrepit, even worse, they were a horrible colour, coral pink. I replaced them with vinyl mainly because I wanted a waterproof floor covering to protect the chipboard floor panels.
If you want a bizzarre use of carpets, my first job, in the mid-80's, was in a new tower block that the was the HQ of a conglomerate, and they had ankle-deep carpets in the men's toilets! There were no urinals, but even so... (The lighting was deliberately extremely dim, one candle would have lit up the whole windowless room more than the electrics did. So if the carpet was manky, you wouldn't have noticed, as it was too dark to see your feet.)