What does Time Machine mean by "Do You Want this computer to claim this backup?"

Time Machine backs up computers. when you change something: computer, logic board, other stuff -- Time Machine notices.

It was asking you if you wanted to claim the backup DRIVE as the official backup destination DRIVE for the new computer.

When you said OK, it began an incremental backup as of that moment, copying changed files onto the existing backup drive you said you wished to claim.

Time machine does not replicate large groups of files, ever.

It is possible enough 'incidental' files, such as new versions of preferences under a new MacOS, and things like that were actually different that it warranted a substantial amount of backing up.

How do I get this set up correctly?

When you set up your new Mac, did you elect to transfer its content from the older one's Time Machine backup? Because that's the way to set it up correctly. Doing that would have populated your new Mac with the older one's content so that you could start using it as though nothing had changed, except you'd have a new Mac running Ventura. Everything else would be the same.

Now you are asking some questions that would not have arisen if you had set things up that way to begin with, and the answers become complicated. Not a problem, but if you really want to do things the right way, erase the new Mac and set it up correctly.

It sounds as though the new Mac created a brand new Time Machine backup, which will remain separate and distinct from the older one's TM backup. Again, not a problem, but an inconvenience. For example if you want to restore backed up item, you will need to ask yourself "do I want to restore it from this Mac, or the other one?"

Since the Mac is new, you can avail yourself of Apple's 90 day complimentary phone or chat support, and they'll walk you through it. Start here: Official Apple Support

It seems Time Machine is working properly, but it started creating a brand new set of backups for what it considers a completely different Mac. That's ok as long as that's what you want to do, but it tends to cause confusion for the reasons I explained.

Furthermore, the older Mac's backups will never be deleted, effectively decreasing the available space for newer backups. The only way to make the drive's entire capacity available to TM is to erase it.