Sure, both are quotients of tensor algebras, but if anything, it's the other way around. Geometric algebra is to exterior algebra as inner product spaces are to vector spaces, or as Riemannian manifolds are to smooth manifolds. I'd say that vector spaces are more general than inner product spaces.
> Why start calling things 2-vectors, when you could call them 2-forms and people would know what you were talking about?
Because they're trying to appeal to the kind of physicist that doesn't know what a 2-form is. Besides, bivector is already established terminology, too.
> But how do you take the inner product a·i of a vector and a 2-form?
Yeah, this isn't great, since there are way too many notions of "inner product" in geometric algebra (see The Inner Products of Geometric Algebra: 10.1007/978-1-4612-0089-5_2). But I guess the intent is clear enough.