How do you use your old computer’s time machine to back up your new one?

I’m not quite entirely sure what question you’re asking, so…

…You can use the existing backup from the new computer by “inheriting” it. This is offered when the new Mac first connects to the old backup that it was migrated from.

…You can use the Time Machine backup to migrate to the new computer: Move your content to a new Mac - Apple Support

…You can shelve the old backups for a month or six, until you’re sure you no longer need the data, and then erase and re-use the storage. Possibly as a second parallel Time Machine archive, or otherwise.

…Erase the existing device, and start backups anew from the new Mac.