This is more an extended comment than an answer to the question. The first thing to note is that there are different strenghts of the classification theorem for surfaces. Of course, there are the differentiable, triangulated and topological setting. But even if we choose such a setting, there are two statements one has to prove (at least in one approach):
Every closed surface is isomorphic to a sphere with handles or cross-caps attached.
The isomorphism type only depends on the number of handles and cross-caps attached.
Both the Zip-proof and the Zeeman proof refered to above in the comments only prove the first part and not the second part. The second part is essentially equivalent to the well-definedness of the connected sum. Especially in the topological setting, this is a subtle point, requiring even in dimension 2 a kind of Schönflies theorem (and being a very difficult theorem in dimension 4). Out of frustration about this state of affairs I wrote up a variant of Zeeman's proof (based on a treatment by Thomassen), but including the well-definedness of attaching handles/cross-caps. (It took me a long time to thus actually understand the argument of Zeeman.)
This is not really an answer to the original question, both because this proof is in the triangulated setting and does not avoid the isomorphism of the original question. But I really want to stress the point that well-definedness of attaching handles or connected sum is something one has to prove and not just hide by abuse of notation.
