Why are the identities of distinct groups treated as being the same? While studying group theory I've stumbled on something which I find to be worthy of discussion. Seemingly by convention we have that the intersection of any two groups will always be of order at least one. More specifically, we argue that all groups will by definition contain "the identity" and therefore we have that the intersection of any two groups must always contain (at the very least) the identity.
My question is essentially as follows - is this a convention? Or are there deeper reasons why we treat this as a fact?
For a specific example, we can consider both a group of integers with identity 1 and a group of 2*2 real matrices with the identity being the 2*2 identity matrix. In set theory I would be happy to argue that the intersection of these two sets is empty (because matrices simply aren't numbers). So by what means can a group theorist concluded that the intersection of these two groups is a non-empty set?
Incidentally, I am aware of both the theory of isomorphisms and the proof that the identity of any single group is unique, however I do not see the relevance of these in this particular case.
 A: When people say the intersection of any two groups contains the identity, they are always talking about two subgroups of a single larger group.  In that case, the identity elements of our two groups must be literally the same, since it is the identity of the larger group.
On the other hand, if you just have two unrelated abstract groups, we normally don't talk about their intersection at all, and if we did talk about their intersection, we wouldn't say their intersection must be nonempty, since as you note, their identity elements may be different.
(In other words, the convention you are asking about does not actually exist.)
A: Usually we take the intersection of two subgroups $A$ and $B$ in a group $G$. Then, by the definition of a subgroup, $e_A=e_B=e_G$ for the identity element. For example, we may consider $A=\mathbb{Z}$ as subgroup of $B=M_2(\mathbb{R})$, with the embedding
$$
n\mapsto \begin{pmatrix} n & 0 \cr 0 & 1 \end{pmatrix}.
$$
Then the identity matrix is the identity of the subgroup of $M_2(\mathbb{R})$ isomorphic to the integers.
