Time Machine error: Disks have same name

What’s actually tripping Time Machine here isn’t the visible disk name you changed in Finder, it’s the internal volume identity macOS uses under the hood. On macOS, your internal disk is split into a sealed system volume (Macintosh HD) and a data volume (Macintosh HD – Data), plus APFS snapshots.

If at any point you cloned, restored, or reused a backup disk that still contains old APFS snapshots or a leftover Time Machine destination with the same volume UUID, Time Machine sees two sources that look identical even though they don’t to you. That’s why wiping files or renaming volumes doesn’t help.

The fix is to fully remove the old Time Machine association and snapshots, disconnect the backup disk, run tmutil listlocalsnapshots / and delete any stale snapshots, then in Disk Utility show All Devices and completely erase the backup disk at the device level (not just the volume) so it gets a new APFS container and UUID.

After reconnecting, re-add it in Time Machine settings. If that clears the conflict, you’ll know the issue was metadata and snapshots, not shared files or something you can safely delete manually.

Although the error you posted indicates something is amiss with disk names, what we see here in your pic of Disk Utility is typical and appears to be in order.

Since we cannot see what your OS sees, you can help us by posting a screenshot of what Terminal can see regarding your mounted drives.

Please open the Terminal app and use this command: diskutil list

The result will be something similar to this:

Also, you might try running the TM backup while booted in safe mode.

How to use safe mode on your Mac - Apple Support 

Safe mode clears caches and keeps the Mac from running certain third-party mods at startup, thus preventing interference with the OS and its smooth operation.

The Macintosh HD Snapshot Volume is the Volume with the Operating System is located

The macOS operating system resides in a sealed and read-only volume that can’t be opened by users or third-party applications

The Macintosh HD- Data Volumes it where the User Account ( Home folder ) located

Time Machine Backup since about macOS 10.15 Catalina will only backup the User Account Data and not contain and full and completed working copy of the Operating System