I think that if we are talking about Grothendieck, we mean implicitly categorical language. In that sense, it is fair to say that Grothendieck introduced the notion in categorical foundations. I was of the supposition that it was Tarski who introduced the notion before, but you clarified its introduction in set-theoretical foundations. The question goes in the direction of Grothendieck' seminar and in that way into categorical foundations. Thanks for your clarification, nevertheless.

@The reason nLab was not enough was that the notion is well motivated, but I would like to know about the particular choice of axioms. You can motivate a notion, for example, a group (resp. topological space) as a theory of symmetry (with localization) but the choice of axioms seems to be part of an experiment (Hausdorff axioms, open subsets,...). I am interested in a motivation for the choice of axioms.

In a letter to R Brown of 15.2.1983 (agrothendieck.github.io/divers/GBCorr.pdf) Grothendieck says: "His thesis should be more than fifteen years ago – I don’t know the reference (just happened to be there when he had his soutenance) nor the present whereabouts of Bénabou." This gives a hint that Grothendieck was aware of Bénabou's work in the 60s.

In "Introduction to bicategories" Bénabou cites "Catégories relatives" of 1965 as an early resource. But in the text relative categories appears as an example of 2-categories (Th 5). I would venture to say that 2-categories appear either in "Catégories avec multiplications. C. R. Acad. Sc. Paris (1963 )" or "Algèbre élémentaire dans les catégories avec multipliplication. C. R. Acad. Sc. Paris (1964)". But I was unable to find these references. Those years appear reasonable to me, the same as those of the SGA 4 seminar.

Thanks, @varkor for your efforts. In the cited reference of Monique Hakim we read: "La notion de 2-catégorie est due à J. Benabou". But the bibliography does not cite any of his works.

Many thanks for collecting this up, but I second @varkor to know who introduced 2-categories.