Wikipedia seems to have an answer
"The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was discovered in 1968 by Jean Bénabou."
But this answer seems to me a little bit odd.
SGA 4 of Grothendieck was held between 1963-1964. There we find the following remarks
"C’est le fait que les 𝒰-topos (éléments d’un univers 𝒱) forment une 2-catégorie, et non plus seulement une catégorie ordinaire comme les espaces topologiques ordinaires, qui constitue du point de vue technique la différence la plus importante entre la théorie des topos et celle des espaces topologiques. Ce fait est la source de certaines complications techniques auxquelles on a déjà fait allusion, mais aussi de faits essentiellement nouveaux par rapport à la topologie traditionnelle."
and yes, I know that SGA 4 was edited and published then in 1972 when the notion was already known. But there is another remark
"Le fait que les 𝒰-topos (éléments d’un univers 𝒱) forment une 2-catégorie permet en particulier de définir la notion d’équivalence de deux 𝒰-topos 𝐸, 𝐸′."
And it is very likely that a notion of equivalence of topoi was already in the years 63, 64 in the original seminars (this is a speculative remark).
SGA4 cites the thesis of Monike Hakim as a former reference but that thesis was published later.
Do we have some conclusive information to add to the hypothesis that was Grothendieck who introduced the notion of 2-categories?
We can also say a few words about the use of Cat, already in use in SGA 1 and suggested even in the Tohoku paper. Grothendieck wrote a paper related to Quillen in 1968 where it is already present the theory of n-categories.
If you think Wikipedia is right, can you please let me know how Grothendieck enters in this picture?