For groups, the "free product" can be taken "generator-wise" and "relator-wise" as done here:
https://ncatlab.org/nlab/show/free+product+of+groups
It is also the case that the "free product" is the coproduct in the category of groups. For other (algebraic, and hence concrete) categories, we can find free objects and write down arbitrary objects in terms of generators and relators of a presentation that specifies them (monoids, quandles, rings, etc.). These categories have somewhat different coproducts, however (coproduct in Ab is not disjoint union, but would, in some sense, be equivalent to some general coproduct were we to instinctively include the commutator as a relator).
I am asking "does this discrepancy matter"? Or is it the case that, whenever we have free objects and coequalizers (and hence coproducts) (as we do in algebraic categories) we will be able to use these coproducts as though they were the "free product" of groups ("generator-wise" and "relator-wise")? Do we simply have to keep track of "category specific relations", like needing to include the commutator for abelian groups?
The question's not crystal clear, but a relevant point is that you can always give the coproduct of two algebraic gadgets given by presentations $A=\langle a_i|r_i\rangle,B=\langle b_j|s_j\rangle$ by "juxtaposing presentations", to get $\langle a_i,b_j|r_i,s_j\rangle$. Thus the coproduct is always the free gadget generated by generators of $A$ and $B$ subject to the relations coming from $A$ and from $B$. Of course in different algebraic theories, there are different relations that always hold, but of course there are also different operations! All this is probably best thought of as being encoded in the "free gadget on some generators" construction, as well as in the coequalizer determined by the relations.