Using old Time Machine external backup drive on a new Mac

After downloading all files from my old mac to new mac, shall I just erase my old external time machine drive and start over? When identifying the old drive as the new time machine drive, I was not asked if I want to erase or claim existing backups, and it seems to be starting over, adding all files on the new computer as new to the old backups. Or is my new mac recognizing the old files already?

Uncertain, would I be best to just erase the old drive and start over on it for my new mac time machine?