We know two finite groups with the same character table might not be isomorphic (e.g. $D_4$ and $Q_8$), but the sizes of their abelianizations are equal (in fact equal to the number of linear characters, which can be read off the character table).
Can we also say the abelianizations of these groups are isomorphic, or is this not necessarily the case?
Yes. The $1$-dimensional characters form a group under pointwise multiplication, and the abelianization is (noncanonically) isomorphic to this group. (It is canonically isomorphic to the Pontryagin dual of this group.)