They are called algae because it is easy to cubbyhole something that way. People feel better when they can put a name to something, that way they feel like they know about it.
Algae grow mostly underwater and do not have a plant body. Red algae grow to be huge (15m) plant-like organisms with modifications that look like stems and leaves. Single celled "green algae" like Chlorella can grow in a mat on the side of a tree.
What this tells you: "algae" is a common non-scientific group term, like ornamental shrub or chaparral. It has little to do with taxonomy and everything to do with the history of language. Putting red or or green or blue-green in front of the word algae is slightly better, but still seldom scientific.
If you read the link it discusses things in detail which is not feasible here on PF.
https://en.wikipedia.org/wiki/Algae