the class of all finite groups is not closed under produtcs - example: the product over all finite cyclic groups - thus it is not a variety of algebras, ie, it's not axiomatizable by universal sentences.
in fact more is true: it's not axiomatizable at all, by a consequence of the compactness theorem for first order theories, namely, that a theory having models of arbitrarily large finite size - again, take the cyclic groups - must have an infinite model.
proofs of theorems pertaining to finite but not to arbitrary groups have more of an 'arithmetical flavor' than a 'general nonsensical' one, and it's not hard to envision the subject being developed inside first-order number theory.
so what is it that makes the subject of finite groups 'algebraic'? why does it appear on, say, syllabi on undergrad and graduate algebra?