> Could the original vendor pass some value to use in the integrity check to prevent replay attacks, and could that value itself encode your personal information?
Well that value is most likely a cryptographic signature, a "challenge" or a combination of both. Unless there's some separate payload you can't really hide arbitrary data in hashes/signatures that would be used in such a process.
In the end "could" is a very loose word, PII as such is not really part of the process. In this current (Apple's PAT) case, the information is "you have an Apple device", can't currently hide anything else in that.