Yes NTFS does that but exFAT, or any other FAT variant I know of, not (unless your tz is UTC). See also: exFAT file system specification.
From exFAT Timestamp Behavior Associated with Different Operating Systems:
But for the moment the summary result is that no tool will reliably display the exFAT timestamps associated with the various tested operating systems. The tools all failed to accurately display the date an action was performed in one way or another for at least one of the operating systems. Importantly this isn’t always a failure of the tool, as above the the differences in implementation of exFAT cause inconsistencies which no tool could account for.
The tools which failed the worst are those which asked the examiner to specify the source timezone, commonly these ignored the provided timezone data altogether and then adjusted all the decoded MSDOS timestamps at the whim of the user. This scenario is most challenging when an item of removable storage media has been written to by multiple different operating systems or multiple instances of the same OS but with different timezone settings configured.