This guy is entitled to use the word "plant" as he will, but it doesn't agree with modern systematics. For example he calls "kelp" a plant, but it is taxonomically closer to the parasite that causes malaria than it is to land plants.
"Macroalgae" is a multi-phyletic category, including eukaryotes of the Archaeplastida group that includes red algae and green algae and the land plants that evolved from green algae, and of the super-group Chromalveolata that includes red tides, brown algae (such as kelp or Plasmodium). Green algae and land plants are grouped together under the kingdom "Plantae" in modern taxonomies.
So "seaweed" as a category includes organisms which are (cladistically speaking) closely related to land plants (green algae like sea grapes or sea lettuce), middling-related (red algae like nori or carageenan) and not very closely related at all (brown algae like kombu/kelp). Of course all organisms are presumably related to some degree.
The seaweed in question is a kind of dulse, a red algae. It's more closely related to land plants than a brown algae like kelp would be, but less related than sea lettuce. Red algae are specifically not included in the Kingdom Plantae. However, layman are free to call whatever they want a plant, even if it's in fact something else entirely, the way they call any small arthropod a "bug", even through true bugs are one of the 75,000 species in the order Hemiptera (out of over a million insect species).