What I'd do.
First, I'd "wipe clean" the m1 Mini using the "erase all content and settings" option (system preferences/settings).
This will restore it to "out of the box" condition.
Next, just "let it sit there" for a moment, and go on to the next step.
DO NOT go beyond the "choose your language" step. NOT YET.
I'd get an EXTERNAL USB3 drive and connect it to the MacBook Air.
I'd download either SuperDuper or CarbonCopyCloner (both are free to download and use for 30 days -- SuperDuper will be easier for your purposes).
Next, I'd use SD to create a cloned backup on the external drive.
I'd connect the external drive to the NEW Mini.
It's now ready to boot for a brand-new setup, so boot it or continue from "choose your language".
When setup assistant asks if you wish to migrate data, YES YOU DO, so "point the way" for setup assistant to the external drive.
Give SA some time to "digest everything".
SA will present you with a list of stuff to migrate.
I suggest you just migrate everything, so turn SA loose to do its thing.
It will take some time, be patient.
When done, you should see your original login screen as it previously appeared on the MB Air.