Why does Euclid write "Prime numbers are more than any assigned multitude of prime numbers."

You can see Aristotle and Mathematics and Actual infinity.

According to the Aristotelian philosophy, we cannot legitimately "handle" actual infinity; i.e. we have no experience of an infinite "collection" but only of an unlimited iterative process (the potential infinity).

Euclid's statement must be understood in this context : we never have a "complete" infinite set of prime numbers, but we have a procedure that, for a finite collection of prime numbers whatever, can "produce" a new prime which is not in the collection.

There are more prime numbers than any (finite) list of them can contain.

cantor's diagonal argument for the uncountability of the reals follows the same pattern: given any list - even infinitely long - of real numbers, he can prove that there are many elements missing; there are more real numbers than any (even countably infinite) list of them can contain.