I've had the same problem too. Time Machine backups started to fail after upgrade to OS 15.0.1. Both on a network attached storage (NAS) device AND on a Mac Mini as backup destination.
On the Mac to be backed up (MacBook Pro M3, source), I went to Settings, then Time Machine, and removed the destination that was failing. But I could not add it again until I did the following.
Backup disks you can use with Time Machine - Apple Support
For backing up that Mac (source) to another Mac computer (destination): I went to Settings on the destination Mac, selected General, File sharing, added the destination device/folder, and then right clicked on the added device/folder, and selected “Share as a Time Machine backup destination,” then clicked OK.
Then on the Mac I wanted to backup from (source), I saw the external Mac destination for doing the backup. Add the new destination, choose to use the same backup device/folder as before, and let it do the next backup.
However, it failed again on the Mini as destination: "The network disk disconnected from your Mac while backing up".
The NAS I'm using is on an AirPort Extreme (old Apple device with USB connection to a 6TB hard drive). It very recently started failing backups too with the same error message as for the Mini as a destination.
Finally, I am running Bitdefender on both the M3 MacBook Pro and M1 Mini, BUT I have disabled its protection for Time Machine backups since that previously interfered with doing backups over the LAN (several years ago now).
I have not disabled the MacOS firewall in network settings on either the MBP or Mini.