The answer to the first question is no. A. L. S. Corner showed that (quoting from the mathscinet review): `given a positive integer $q$, there exist standard abelian $p$-groups $G$ and $H$ with no elements of infinite height such that $G^n\cong H^n$ if and only if $q$ divides $n$'.

So if $q=2$ we get $G\times G\cong H\times H$ -- and certainly a bijection as required by the question -- even though $G\not\cong H$.

This result is in the paper

A. L. S. Corner On endomorphism rings of primary abelian groups. Quart. J. Math. Oxford Ser. (2) 20 1969 277–296.

(This is basically a duplicate of my answer to If $G \times G \cong H \times H$, then is $G \cong H$? this question in turn was a duplicate of When is $A$ isomorphic to $A^3$?)

The answer to the first question is no. A. L. S. Corner showed that (quoting from the mathscinet review): `given a positive integer $q$, there exist standard abelian $p$-groups $G$ and $H$ with no elements of infinite height such that $G^n\cong H^n$ if and only if $q$ divides $n$'.

So if $q=2$ we get $G\times G\cong H\times H$ -- and certainly a bijection as required by the question -- even though $G\not\cong H$.

This result is in the paper

A. L. S. Corner On endomorphism rings of primary abelian groups. Quart. J. Math. Oxford Ser. (2) 20 1969 277–296.

(This is basically a duplicate of my answer to If $G \times G \cong H \times H$, then is $G \cong H$? this question in turn was a duplicate of When is $A$ isomorphic to $A^3$?)