Part of the problem is that "like" does not have a precise meaning. It depends on the context. I don't know what you know, so I'm gonna go real basic. There are a few things to mention. (First see the other comment about "factors" vs "terms".)

You know how 2x is x+x, and 3x is x+x+x? So the number in front (the **coefficient**) basically counts the thing (**factor**) right after it. But what counts as "the thing" depends on what you care about. If you're focused on x as a **variable**, and b is just some number (known or unknown), you might have 2bx and call "2b" the coefficient. But if b and x are both variables of interest, you might call "2" the coefficient on the "bx" term. These words communicate intent, not mathematical truths.

Now, "like" terms are the ones where the coefficients can be different, but the rest of that term is *exactly* the same. So in your list in your OP, "3b" and "2b" are like terms. I can actually tell without context here because 2 and 3 are never variables. All that's left is b. 1) If b is a variable, then what's left after removing the 2 and 3 is just "b" in both cases, so exactly the same. 2) If b is a constant, then both are constant terms, so they are like. If it helps, you can think of them as (2b)\*1 and (3b)\*1.

I guess that last sentence might be a little confusing since 1 is not a variable. But really the point of identifying "like" terms is to clean up clutter. Just like in life, you group together things in categories, but context might change what categories you care about. Really when you "combine", it means "factor in a minimal way that we all agree is fairly convenient".

Hopefully that at least clears up why it can be confusing (at least without context). NOW, without further ado, without context I would assume in the example in your OP, that a and b are both variables "of interest" (my phrase, not a common one). In that case, only the actually numeral in front can change for two terms to be considered "like". So "3b" and "2b" are like, and "a", "a", and "-2a" are like. The "5b^(2)" has no other like terms, since they would have to be exactly #b^(2).