So I think I came up with a good idea. I bought this and just hung it on the tiles. It's actually meant to be standing next to the kitchen sink, to put the clean cutlery in it after doing the dishes. (Bought it from Lagerhaus.)
It doesn't fill upp with water, saliva and bacteria, it's really easy to clean but it takes a while before even needing it. It doesn't get all those white stains from the toothpaste even though it's black. And the sink below is also easy to clean, just swiping a cloth or splashing some water underneath it sometimes.