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It is a nook/soap dish. The actual soap dish hung on those 2 studs in the bottom front. It looks like the chrome plating is worn off so there is no effective way to clean it. Your replacement options are pretty varied. There are many prefab nooks that can replace it. Finding one the same size may be a challenge but a larger one will work you just have to cut the tile with a grinder to make the hole bigger.
There is a very remote chance that you might find one at an architectural salvage store but that's a real shot in the dark. If you want a soap dish your best bet would be to replace this nook.
Nothing that would specifically identify this particular item. It's a "vintage" partially recessed soap dish.
It's highly unlikely you'll find one anywhere.
If you want me to search one out for you I will require some additional compensation since it will be very time consuming and I can't guarantee I will be successful in my search.
Sorry I didn't have better news.