This is a side matter to the main question here, but I wanted to add a bit more on the history of the universe concept, since this seems to be less widely known than it deserves.

Namely, universes were introduced and analyzed by Zermelo in his 1930 paper, several decades before Grothendieck:

• Zermelo, E., Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre., Fundamenta 16, 29-47 (1930). ZBL56.0082.02. (translated in Ewald, William Bragg, From Kant to Hilbert. A source book in the foundations of mathematics. Two volume set., Oxford: Oxford University Press. 1440 p. (2000). ZBL0935.01002.

To be sure, universes are the central focus of this paper, in which Zermelo defines the universe concept, considering them as set-theoretic realms for mathematics. His version of the universe concept allows for (but does not insist upon) a set of urelements. He proves his famous quasi-categoricity result, establishing that the universes are precisely the models of second-order set theory $\text{ZFC}_2$, and that they are linearly ordered and connected with the inaccessible cardinals of Hausdorff, and he analysizes their automorphism groups, which are induced by permutations of the urelements, if any. In addition, he considers various philosophical aspects of moving from one universe to the next, in that various proper classes become sets in the next universe, which is a central use case in category theory.

Zermelo also explicitly considers a version of the universe axiom:

...the existence of an unbounded sequence of boundary numbers [heights of universes, or inaccessible cardinals] must be postulated as a new axiom of 'meta-set theory', and in so doing the 'consistency' question must be looked at more closely.

Zermelo's 1930 analysis thus seems in several respects to surpass Grothendieck's later analysis, which to my knowledge does not provide the categoricity result and does not engage with the consistency issue.

In light of their origin in Zermelo's work, the Grothendieck universes are now also known as Zermelo-Grothendieck universes.

This is a side matter to the main question here, but I wanted to add a bit more on the history of the universe concept, since this seems to be less widely known than it deserves.

Namely, universes were introduced and analyzed by Zermelo in his 1930 paper, several decades before Grothendieck:

• Zermelo, E., Über Grenzzahlen und Mengenbereiche. Neue Untersuchungen über die Grundlagen der Mengenlehre., Fundamenta 16, 29-47 (1930). ZBL56.0082.02, eudml. (translated in Ewald, William Bragg, From Kant to Hilbert. A source book in the foundations of mathematics. Two volume set., Oxford: Oxford University Press. 1440 p. (2000). ZBL0935.01002.

To be sure, universes are the central focus of this paper, in which Zermelo defines the universe concept, considering them as set-theoretic realms for mathematics. His version of the universe concept allows for (but does not insist upon) a set of urelements. He proves his famous quasi-categoricity result, establishing that the universes are precisely the models of second-order set theory $\text{ZFC}_2$, and that they are linearly ordered and connected with the inaccessible cardinals of Hausdorff, and he analysizes their automorphism groups, which are induced by permutations of the urelements, if any. In addition, he considers various philosophical aspects of moving from one universe to the next, in that various proper classes become sets in the next universe, which is a central use case in category theory.

Zermelo also explicitly considers a version of the universe axiom:

...the existence of an unbounded sequence of boundary numbers [heights of universes, or inaccessible cardinals] must be postulated as a new axiom of 'meta-set theory', and in so doing the 'consistency' question must be looked at more closely.

Zermelo's 1930 analysis thus seems in several respects to surpass Grothendieck's later analysis, which to my knowledge does not provide the categoricity result and does not engage with the consistency issue.

In light of their origin in Zermelo's work, the Grothendieck universes are now also known as Zermelo-Grothendieck universes.