Time capsule backup not recgonized after system restore

Is there an easy way in Sierra to delete the backups that I no longer need?

it's easy if it works.

Open any Finder window and look for the name of your Time Capsule either under the Devices or Locations heading on the left side of the Finder window

Click on the Time Capsule name and another window will open to either display the name of the Time Capsule hard drive.......named "Data" by default.......

If you see a Connect button, click on it to connect to the hard drive. You may be asked for the Time Capsule admin password.

If you have previously connected to the hard drive this way, "Data" will be displayed in the window

Double click on "Data" to open the drive

You will see sparsebundle file with the name of the Mac.....which contains all the backups for that Mac

Make sure that you have the right file to delete

Click on the file to highlight it, then click the gear icon at the top of the window and click Move to Trash

If things go well, it will take 5-10 minutes or so on average to delete the file

"The identity of the backup disk “Airport Time Capsule” has changed since the previous backup" and my computer wants to do a full backup which is not necessary.

If you "restored" your Mac, you have in effect a "new" Mac identification wise, so Time Machine does not recognize your "new" computer.

The backups that you have now on the Time Capsule were from your "old" Mac and Time Machine will not back up to that file. You'll have to go with what Time Machine wants to do......start a new backup of your "new" Mac and work forward from that point.

A secondary question I have is that there is an old backup on my time capsule from my previous computer, which I definitely do not need anymore. Is there a way to delete that?

Apple changed a ton of stuff with Time Machine in recent operating systems like Catalina and Big Sur, so this may or may not be possible. You can always erase the Time Capsule drive to wipe everything off and start again if that would work, which will only take a few minutes.

If that is not acceptable, we can try to go through the steps to erase only the backup(s) that you don't need anymore, but there are no guarantees on success with the newer operating systems. Recently, using my Mac running Big Sur, I tried to delete an old Time Machine file on the Time Capsule that I no longer needed. Nothing but error messages when I tried this multiple times.