Abbot & Costello "Who's on First" with a pragmatic (or other linguistic) perspective?

I would say that the primary thing going on in that routine is the break down of 'conversation repair'. The two interlocutors simply repeat the same conversation repair approach and do not learn from its failure. The constant repetition is what makes it funny.

It is also interesting how through constant repetition, the players make the rather implausible notion of players being nick named 'Who', 'What', 'I don't know', 'Why', etc.

Of course, once those elements are in place, you're dealing with structural and lexical ambiguity. This particularly well illustrated in this sequence where Costello tries to initiate conversation repair:

Costello: When you pay off the first baseman every month, who gets the money?

Abbott: Every dollar of it. And why not, the man's entitled to it.

Costello: Who is?

Abbott: Yes.

Costello: So who gets it?

Abbott: Why shouldn't he? Sometimes his wife comes down and collects it.

Costello: Who's wife?

Abbott: Yes. After all, the man earns it.

Costello: Who does?

Abbott: Absolutely.

For example, in 'Who does?' the question has a completely different interpretation if 'Who' is treated as a proper noun or a pronoun. If it's a pronoun, the answer 'Absolutely' makes absolutely no sense. However, it's perfectly acceptable if 'Who' is a noun.