Can you find the fallacy in this inductive argument that all things are the same?

In honor of the start of the school year, here is what I think is a fairly interesting use, or rather misuse, of mathematical induction. When I first saw this problem, it took me a while to catch the flaw.

To prove: Everything is the same as everything else.
Base case: n =1. Obviously everything is the same as itself.

Inductive step: Assume that any n things are the same as each other. Choose any group of n+1 objects. The first n are all the same by the inductive assumption. So are the last n. Since the two groups overlap, all n+1 objects must be the same.

We conclude that any number of objects are all the same, that is, everything is the same as everything else.