User blog comment:Deedlit11/Extending the fast-growing hierarchy to nonrecursive ordinals/@comment-30754445-20170419125319/@comment-1605058-20170419153307

It was Kleene himself whoh has used exponentiation there and using these two functions has since became customary.

Kleene's original paper doesn't justify the choice in any way, but my guess, based on the passage "The system \(S_1\) [i.e. Kleene's O] is modeled after the system of formulas in the \(\lambda\)-notation assigned to represent ordinals by Church and Kleene.", is that the mentioned system of notations has required one to use infinitely many functions (perhaps since it has used functions of arbitrary arity) for which the use of prime factorizations is perfect, and Kleene has decided to keep similar notation.

You are right, however, that any two injective functions with disjoint images will work just as fine.