These two groups have the same order. Also, we cannot show that the groups are not isomorphic by comparing the order of groups' elements. Thus, it seems that these two groups are isomorphic.
How to prove that $\mathbb{Z}_{84} \oplus \mathbb{Z}_{72}$ is isomorphic to $\mathbb{Z}_{36} \oplus \mathbb{Z}_{168}$.
They are indeed isomorphic, and to notice this, use the fact that $\mathbb{Z}_{mn} = \mathbb{Z}_n \times \mathbb{Z}_m$ if and only if $\text{gcd}(m,n) = 1$
And since $84 = 12 \cdot 7$ and $72 = 8 \cdot9$ you may decompose the L.H.S into
$\mathbb{Z}_9 \times \mathbb{Z}_8 \times \mathbb{Z}_7 \times \mathbb{Z}_3 \times \mathbb{Z}_4$ and show that something similar may be done with the RHS.
More generally,
$\Bbb Z_m \times \Bbb Z_n \cong \Bbb Z_{\gcd(m,n)} \times \Bbb Z_{\operatorname{lcm}(m,n)}$
(see this question and this question)
Therefore, $$ \mathbb{Z}_{84} \oplus \mathbb{Z}_{72} \cong \mathbb{Z}_{12} \oplus \mathbb{Z}_{504} \cong \mathbb{Z}_{36} \oplus \mathbb{Z}_{168} $$ because $\gcd(84,72)=12=\gcd(36,108)$ and $\operatorname{lcm}(84,72)=504=\operatorname{lcm}(36,108)$.
