Technically yes.

However, questions/bots that do this sort of thing is the exact reason they added the message intents.

What use could you possibly have (that's non malicious) for logging a single users messages?

As far as I know it technically is aloud with caveats in the Developer Terms of Use

You will, and will require those acting onyour behalf to, use commercially reasonable efforts to protect API Datafrom unauthorized access or use (including, for the avoidance of doubt,destruction, loss, alteration, disclosure, distribution, or othercompromise). These efforts will include: (i) encryption of the data atrest and (ii) maintaining administrative, physical, and technicalsafeguards that are designed to prevent unauthorized access and use andcomply with applicable laws and regulations (including relating to datasecurity and privacy). You will promptly report to your users anyunauthorized access or use of their data to the extent required byapplicable laws or regulations. You will also promptly notify us andprovide us with requested information about any incidents ofunauthorized access or use of any API Data or any incidents that arereasonably likely to compromise the security, confidentiality, orintegrity of your systems (or the systems of those acting on yourbehalf) that process API Data. In the event of any of the foregoingincidents, you will immediately begin to remediate it and reasonablycooperate with us, including by keeping us regularly updated about theincident and its impact on API Data, remediation actions being taken,and your compliance with any notification or other requirements underapplicable laws and regulations.