Are seaweed plants

Yes,

Seaweed is a loose colloquial term encompassing macroscopic, multicellular, benthic marine algae.[1] The term includes some members of the red, brown andgreen algae. Seaweeds can also be classified by use (as food, medicine, fertilizer, industrial, etc.).

A seaweed may belong to one of several groups of multicellular algae: the red algae, green algae, and brown algae. As these three groups are not thought to have a common multicellular ancestor, the seaweeds are a polyphyletic group. In addition, some tuft-forming bluegreen algae (Cyanobacteria) are sometimes considered as seaweeds - "seaweed" is a colloquial term and lacks a formal definition.