Is 'to gain an advantage' a pleonasm?

The definition of "advantage" is roughly "something positive". Definitions for "to gain" are rather varied, but usually mean "to win something" with a positive connotation.

With this in mind, is "to gain an advantage" a pleonasm or not? Does "gaining" imply "advantage"?

Examples of definitions of "to gain": 1 and 2.

No, because a pleonasm needs to not just repeat something, but to use more than is necessary.

As the first dictionary definition you link to says, the sense of gain you are talking about is transitive. As such it must have an object; you cannot just say "he gained" in this sense of the word (you can in others).

You could use another noun or noun phrase as the object, but which? You could have a particular noun that is more specific:

Repeatedly the Germans and Russians gained control of the high ground outside Stalingrad from each other.

That would be better than just "an advantage" in most cases, though in some you might still need to point out that the more specific noun did indeed bring an advantage with it. (Or if the context already explained that you were talking about the fight to control Mamayev Kurgan during the Battle of Stalingrad, then repeating that point would be an even worse repetition than just saying "...gained the advantage").

Otherwise you are left with vagueness:

He gained something.

Gained what? Who cares?

As such, "to gain an advantage" is often reasonable.

Jon Hanna's analysis is valid, but I think a simpler angle suffices to refute the notion that "to gain an advantage" is a pleonasm. Namely, the fact that gain even has some senses that are not strictly positive means that you need context to even know which sense of gain was used. As such, advantage gives you that context, and is thus not redundant.