Does it have threads on the spout? Because you could hook a hose to it. Or you could get it replumbed to go outside as another hose bib. Unless you have a waste pipe, the only exit is the floor drain.

Also, you could tile over the drywall under the faucet. Or put in a cheap utility sink with a "drain" that's just a pipe down to the floor.

Its probably just for filling a bucket or attaching a hose to it. A long time ago I lived in a house that had one like what you describe and we hardly ever used it. Once in a while to fill a bucket with water and that was about it.

We have a hot/cold bib installed like that. We chose that method instead of a utility sink. Less chance of freezing the works in the case someone left the garage door open.
It comes in pretty handy for washing the dog or for the outdoor shower with muddy kids.
I do have a 2 gallon square bucket that I leave under the faucet to catch extra water those frost proof faucets let out.
Our drywall is textured, painted and trimmed. I haven't had much trouble with it getting wet. When it does I just wipe it down with a micro fiber cloth.

Peace