Seaweeds are commercially farmed around the world. Korean "nori" (or "gim") export amounts to $600M per year.

> Are there any “domesticated” marine plants?

A few. Marine upper plants are very rare, but red algae are harvested. There are several cultured marine Protists (I wouldn't say domesticated, but this is debatable) and also aquatic Cyanobacteria.

China grows most of them. Japan or Korea also.

Interesting question. They are called Sea Vegetables. But they do need seawater. Now seawater aquariums are a thing so it should be manageable to "farm" those.