Glue the wallplate to the tiles using a construction-type glue (like Pattex Kleben statt Bohren) and allow it to cure properly (usually 24h). You can then hang the tray into it and place your body care product into it. For ease of cleaning you can simply lift the tray out of the wallplate.

This looks a bit flimsy but it is surprisingly strong. I wanted a slim design and fully expected them to break, after which I would have to redesign them, but they hold up quite well. I have broken a couple, but due to excessive force (mainly by inadvertently knocking them off the wall etc.) - but you can always print another one.