Under Monterey, Time Machine drives show as "you can only Read" in get info. Mine look that way also.
Apple's knowledge base warns that under Monterey the entire drive is taken over for Time Machine. It is recommended to dedicate the entire physical drive for Time Machine but one can also partition it and dedicate a partition to Time Machine if you want to use it also for other storage. If you are sharing the same partition for Time Machine and other misc storage, that could create a problem and unpredictable results might occur.
Also, after going from Catalina to Monterey, the first backup is typically very large and takes a lot of time (that was my experience) because many things are changed under Monterey. It is possible that Time Machine calculates that it needs more room than there is available? That calculation allows for lots of margin, so one needs to have a lot of additional space for the backup to proceed even if in the end it turns out not to need it.
You did not indicate the amount of free space on the backup drive or what model you are using. Also, if you are using any third party disk firmware or software disk tools/utilities, those need to be compatible with Monterey or operations like Time Machine may fail even if they worked under earlier MacOS. In addition, anti-virus, security, Mac "cleaner" software, and security tools/suites can all interfere with normal MacOS operations. Are you using any of the above items?
When I went from Catalina to Monterey I was using two Time Machine drives, both 100% dedicated to Time Machine, both were HFS+ when used under Catalina. I reformatted one to APFS and use it for Monterey Time Machine, the other I left as is (HFS+ with a long history of backups) and while the first Monterey backup on that drive took a very long time (longer even than the first backup on the freshly erased APFS drive), subsequent backups have been quite fast. The two drives I am using for Time Machine with this Mac (16-inch Macbook Pro) are a G-Technology SSD (APFS) and a Western Digital SSD (HFS+).
