My guess is that people (because of similar mind architecture) share beliefs (persuadability) in statements of a stronger system, of which PA and ZF only capture a rough outline. This shared theory accounts for platonism, or detail of "standard model". Capturing the whole of this theory, or even the whole of standard model of arithmetic, is on the same order of difficulty as formally defining human preference (because people don't have reliable access to those intuitions). Part of progress in mathematics consists in understanding this shared theory one aspect at a time. Other part is development of tools given formalist position, that is assuming very little and then playing the symbol game, preparing to consider various possibilities, but this doesn't address the question of motivation for entertaining one formal system and not another, that is platonist's intuition.

(It's similar to how we all solve the problem of induction approximately the same way, which means that we share the same "theory of reality": we reach the same conclusions from (enough of) the same observations.)