Isomorphic subgroups such that the quotient is isomorphic.

There even exist nontrivial finitely generated groups $G$ isomorphic to $G\times G$, hence, taking the normal subgroup $H=G\times 1$ in $G\times G\cong G$, we obtain an isomorphism $G\cong G/N$. See

J. Jones, Direct products and the Hopf property, J. Austral. Math. Soc. 17 (1974), 174-196.

Such groups $G$ are necessarily nonabelian. I am not sure about finitely generated torsion groups, but, most likely, they exist as well.

It is unknown if there are nontrivial finitely presented groups $G$ isomorphic to $G\times G$; see

R. Hirshon, Misbehaved direct products. Expo. Math. 20 (2002), no. 4, 365–374.

Under pointwise addition the family of functions $\Bbb F_2^{X}$ from any set $X$ to the finite field $\Bbb F_2$ of two elements is an abelian group all of whose non-trivial elements have finite order 2.

Let $X=\Bbb N$ and let $2\Bbb N, 2\Bbb N+1$ be the set of even/odd positive integers respectively. Let $H$ be the $\Bbb F_2^{\Bbb N}$ subgroup of functions which vanish on $2\Bbb N+1$. Clearly, $$H\approx\Bbb F_2^{2\Bbb N}\approx\Bbb F_2^{2\Bbb N + 1}\approx\frac{\Bbb F_2^{\Bbb N}}{H}$$ and because $\Bbb N,2\Bbb N$ have the same cardinality we also have
$$\Bbb F_2^{\Bbb N}\approx\Bbb F_2^{2\Bbb N}$$

Here, we can replace $\Bbb N$ with any set $X$ and replace $2\Bbb N, 2\Bbb N+1$, respectively, with any two subsets of $X$ that partition $X$ and have the same cardinality as $X$.

Take $G=\mathbb{Z}^{\infty}\times\mathbb{Z}^{\infty}$.