Can NFC tags log data?

No. Most are able to store the number of scans made but that is all they can do.

It is possible to re-encode NFC tags so it would be possible for theme reader device to add or modify data on the NFC tag for the purposes of logging but that ability lies with the phone not the tag.

Additionally the amount of memory in most NFC tags is tiny. In many cases they cant store more than this paragraph. Using an NFC tag to store logging data would therefore be very difficult and somewhat limiting.
Reader - 14 Jan 2023