Back up Mac Air to Mac Mini?
Can I back-up my MacBook Air [osMonterey12.21 to my Mac mini Model Identifier: Macmini9,1 Chip: Apple M1 Total: 8 Memory: 8 GB
Mac mini, macOS 12.1
The only way would be over a network to a drive attached to the Mini and enabling and setting up Time Machine in the correct way on both machines.
However, @hcsitas has already given you the the best, easiest, and most reliable method and a whole lot less fuss than that.