Is it safe to say that linear algebra is "an algebra" and that abstract algebra is another algebra? Is an algebra also a set or something else?
"Algebra" means several things, all of which originate from an Arabic term "al jebr" from somewhere around the 9th century, which means something like "the bringing together of parts".
One modern meaning is that "algebra" is some formal system for manipulating symbols. High school "algebra" meets this meaning. "Linear algebra" also fits this meaning, as does "abstract algebra".
Another meaning is that "an algebra" is a vector space where there is a way to multiply two vectors together to result in another vector. This multiplication need not be commutative or associative, but it does need to conform to distributive laws.
If you've seen something referred to as "an algebra" somewhere, it is most likely not referring to a subject class (like high school algebra or linear algebra) but rather some specific vector space equipped with multiplication.