When Time Machine Backup stops does it display an information message to the effect "can't backup a .... " naming a specific file or folder? This was happening to me both with the automatically scheduled backups and the manual (Backup Now) options. The file or folder it referred to wasn't always the same, but all were in the Users/yourusername/Library/Containers/ folder. Some times it stopped on a Chrome sub-folder, sometimes on a Safari folder, sometimes on a Firefox folder and at the end repeatedly on a Siri folder/file. When I excluded the Siri file (“/Users/yourusername/Library/Containers/com.apple.siri.media-indexer/Data/mediatitlesfinalTable.tdb") backup worked regularly. After the Monterey upgrade to 12.1 the error/warning occured again, except when I clicked on "Ok" backup continued and completed. Subsuquent backups did not have this problem.
If you are getting a different message, copy it or take a screen shot and share it with the advisor you speak to next. Use the same case # so that it will be escilated to a higher level. The first level of support you speak with is usually just basic level support and directs calls/cases to a more technically skilled support level. If they are not aware of this problem ask to be transfered to a Senior Tech support person, or schedule a call back with a Senior Tech. That level will have the experience to understand the issue, access to reports of the issue and can message the OS Engineers with additional information (and get back additional information). If we don't participate in the solution to this problem, it will take longer to be resolved. The OS Engineers can only do so much testing, we are able to provide much more varied information from our different user environments.
I've participated in the problem solving of this problem, and a few others, that my wife thinks Apple should give me their latest top end MacBook Pro and top end iPad Pro. That's never going to happen, and I don't expect anything more than a thank you and the opportunity to help both Apple and those on this community discussion thread. I'm retired and I enjoy the opportunities to be back in the IT problem solving mode.
I'm just an end user, not an Apple employee nor an Apple consultant. I do have a half century of IT experience and this issue with the release of Monterey 12.0 and the partial fix with release 12.1 is something I've experienced numerous times.
I've been told the OS engineers are aware of the problem (since November) and are working on a resolution. Since my backups are working as designed and as I expect, I'm being patient for another update release.
I hope this helps/works for you. Happy New Year!