and right now what is taught in biology classes is that there are 6 kingdoms of life- archae, eubacteria, animalia, plantae, protista, and fungi. Algae fall under the protista kingdom because they don't have REAL leaves (with cuticles, layers, etc), no REAL stems, no REAL (nutrient absorbing) roots, etc. and many are unicellular anyway. Not all algae are green plants either... red algae, brown algae, and diatoms aren't green.
yes, plants did evolve from GREEN algae, (did I mention kelp is a brown alga?) but that's the point... they evolved.
Just like protozoans aren't animals, or slime molds fungi, algae aren't plants.