Screen mirroring issue between MacBook Pro and Mac mini

Your 2014 Mac mini is recent enough to support being an AirPlay Receiver for AirPlay to Mac. Your 2015 MacBook Air isn't. This explains why you can use the new Mac mini's screen as an AirPlay screen for the 2015 MacBook Air, but not vice-versa.

See: Continuity features and requirements for Apple devices - Apple Support

You're going to need at least one real hardware monitor for that M4 Mac mini – whether you have an extra AirPlay display or not. There are third-party applications which can let older Macs receive AirPlay video, but AirPlay (any version) is not as good as a real hardware video connection. If you want two displays, do yourself a favor and buy two displays.

That is not a reliable solution and you will be sorry you pursued it if you persist due to its lack of reliability. Just purchase an external display (they start at less than $100) for the MM and move forward. Macs are computers, they are not designed nor intended to be an external display for another device, yes including another Mac.

.

I am pretty certain that if you do not just get an external display for the MM you eventually will if for no other reason than frustration.