transferring external hard drive / time machine to a new device without using the back up?

I have bought a new macbook air 13" M1 to replace both an old macbook air and a newer Mac Mini. I have used migration assistant to move the contents of the old macbook air to the new macbook air, and I have cleaned up and manually moved documents from the Mac Mini to the new macbook air. I am now at the point of looking to factory reset the Mac Mini, but I need to move the seagate expansion drive to the new macbook air. Can I just unplug it and move it across? I am assuming there will be prompts to stop it from migrating a back up that I do not want. Or do I need to 'wipe it'? I'm a bit non tech and the time machine was initially set up by a tech person.

Also because it will then be used with a macbook, (is/was permanently attached to the Mac Mini and did auto backups) do I just plug and unplug as needed and it will just back up whenever plugged in?