"A category $\mathfrak{C}$ is the data of the collection $\textrm{Obj}(\mathfrak{C})$ of objects of $\mathfrak{C}$ and for each objects $X,Y$ of a set of morphisms $\textrm{Hom}_{\mathfrak{C}} (X,Y)$, such that ..." A perfectly fine classic where one informally speaks about collections of objects.
First example, the category $S$ of sets. Then whatever kind of "construction" $\textrm{Obj}(S)$ may be, it is for sure not a set, as "the set of all sets does not exist". Another perfectly fine (Russell's) classic in naive set theory, "solved" in ZFC.
I never thought about which set theory I was in when I was considering cathegory theory or doing algebraic geometry (even if Bourbaki's appendix in SGA IV 1 and some "non-boundedness" questions around flat topologies puzzled me) but apparently I was in ZFC. But in ZFC, I don't know what a "collection" is.
What are the reactions (common or not) to this "issue" ?
You should read Mike Shulman's Set theory for category theory. It gives lots of ways to tackle the issue and compares them (their strenghts and weaknesses). This allows you to 1- make sure that what you do is valid in some/all of these formalizations, 2-pick your favorite solution if you have one and stick to it and maybe 3- forget about it if you only want to know you're on solid ground (no matter what the ground is)