Depends on what you call a plate. We definitely eat soups from "talíř" ("hluboký talíř") here. Can you look up some pictures of that and tell us whether you would call some of that a plate or everything a bowl?
However, "bowl" is accepted as an answer.
At least in the US, these are most commonly called "bowls," or, less commonly, "soup plates." They would be used for eating soup, cereal, sometimes salad, and other "messy" foods. The flat ones, or ones with a very slightly raised edge, are the ones called "plates."
But would you duplicate "soup" as in "His soup plate is full of soup." or would you just use "His plate is full of soup." (provided you would want to call it a soup plate instead of a bowl)?
This is a good question, but one that I can't answer definitively.
In my day-to-day life I've never actually heard anyone refer to bowls as "soup plates," other than in stores that sell expensive dinnerware or in posh restaurants! But "His plate is full of soup" would be strange. People just wouldn't, at least in my experience, associate soup with a plate... it could escape too easily! :-)
Jeho talíř je naplněn polévkou. Maybe it could be accepted as well but it is not the closest translation.
It is genitiv singular here. But nominative plural has the same form.