From the wikipedia article on the Skolem paradox:
A central goal of early research into set theory was to find a first order axiomatisation for set theory which was categorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem's result showed this is not possible, creating doubts about the use of set theory as a foundation of mathematics.
I would like to know if the term "categorical" for that property here lead to the naming of category theory. Maybe because all the problems of non-absoluteness don't happen there? How is the relation?
This is not the same category as in category theory.
This is a model theoretic concept of categoricity. We say that a theory is categorical if it has exactly one model up to isomorphism. We know that a first-order theory is categorical if and only if its only model is finite.
So for first-order theories we have a weaker notion, $\kappa$-categorical, namely all models of cardinality $\kappa$ are isomorphic.
Skolem's paradox showed that ZFC is not categorical, in the model-theoretical sense of the word, although historically I am not sure if the term was coined at the time.
On the other hand, category theory deals with notions of categorizing mathematical objects, "groups" or "compact topological rings".