I received my MacBook Pro back from repair (got a new Logic Board which includes the SSD) and restored from my Time Machine backup. I made sure to restore everything, including all settings. Now I was trying to do my first Time Machine backup only to realize that instead of inheriting the history from the Time Machine it had just restored from, it started to do a new full backup. Apparently, while everything had been properly restored, the device name had not, and I realized that too late, so instead of adding the new backup to the backup history of my existing device, it added a new device just called "MacBook Pro".
I don't have that much space on my external hard drive Time Machine so I can't just waste an entire 900 GB for another full backup for the same files that it had just copied from that same backup disk.
So, I stopped the backup, deleted it, renamed the computer to the old name, and also made sure it inherits the history by telling it
sudo tmutil inheritbackup <path to previous backup>
(as other posts on this site suggest).
However, it still tries to do a full backup. I.e., it still seems to not recognize the fact that essentially all files (except for a few I changed yesterday) are exactly the same files already baced up. It seems now I'm wasting 900 GB of space on my external harddrive for what actually should just be 1 or 2 GB.
After all, I did let it it finish the backup, hoping that it would nonetheless deduplicate or overwrite the files during the backup. Unfortunately, it did indeed to another full backup; there are now 900GB less frees space on the external backup harddrive. I'm still curious, for next time, how can I make sure that when reinstalling from a TimeMachine backup, that the newly installed system is then also associated with that backup?