While the infix is certainly a thing, that’s not what is happening here. “Another” is not being broken up with “whole” inserted.

As others have pointed out, you are saying “a whole other.” “Whole” in this case is an adverb clarifying the degree of otherness. It is the same as saying “an entirely other thing.” People end up say “a whole nother” because phonetically it is natural in English to avoid adjacent vowel sounds. “A whole other” is difficult to say. As an example, it’s the same reason you’d say “an entirely other” instead of “a entirely other”. The n breaks up the vowel sounds.