What is the intuition behind the direct product and the direct sum of groups being identical for finite groups?

Question

I have seen the statement that the direct product of groups $G_1\otimes G_2 \otimes … \otimes G_N$ is identical to the direct sum of groups $G_1\oplus G_2 \oplus … \oplus G_N$ for finite $N$.

What is the intuition about this being true?

Answer 1

It's basically the fact that finite coproducts and products coincide in the category of Abelian groups.

There's a deep connection of these coincidences with addition, as discussed here in details.
Briefly, for any category with zero morphisms:

There (naturally) is a commutative monoid structure ('addition') on the set of morphisms $A\to B$ for any objects $A,B\ $ iff$\ $ finite coproducts and products coincide

Now, pointwise addition of homomorphisms $f,g:A\to B$ produces again a homomorphism $A\to B\ $ (if $B$ is commutative), and thus it gives rise to a commutative monoid structure (with inverses, i.e. an Abelian group).