What is the official science name for seaweed extract fertilizer?

I'm not sure there is a single answer to your question. At least from a regulatory perspective, the definition of "fertilizer" varies from state to state. In some states, seaweed extract, or kelp extract as it may also be called, may not contain enough nutrients to meet their legal definition of fertilizer. In others it does.

The Association of American Plant Food Control Officials (AAPFCO) is the closest thing there is to standardization of fertilizer regulations. In their official definitions, AAPFCO only acknowledges the term "kelp," which it defines as "the dried marine algae fo the botanical divisions of Rhodophyta (red algae), Phaeophyta (brown algae), and Chlorophyta (green algae)." If a product met the definition of "fertilizer," it's official name for AAPFCO purposes would be "kelp extract fertilizer."

Although several seaweed species are used in horticultural products, the most common is Ascophyllum nodosum.