All of these numbered universes live inside of a top-most universe which contains all the types and universes. You can't quantify over the topmost universe, for exactly the same reason you can't form the set of all sets in ordinary set theory.
If you do pencil-and-paper mathematics particularly pedantically, when you define something like id, you will say you are working in Tarski-Grothendieck set theory, and then be careful to call it a "definition schema", rather than a "definition", because you are "really" giving a family of definitions, once for each Grothendieck universe. Distinguishing schemas and definitions is a little inconvenient on a computer -- the naive implementation forces you to implement everything twice -- and so one extra top universe is introduced as a home for all of the smaller universes. (Though Mizar apparently chooses to support universe-generic definitions, and to explain them to users as schemas.)
IME, these issues are generally ignorable, both on paper and on computers. Harper and Pollack gave an algorithm for inferring universe levels in the early 90s, and apparently it worked so well that they were worried their implementation was wrong (ie, would always say OK). So they carefully coded up various set-theoretic paradoxes in type theory and were relieved to discover their implementation rejected them.