Adapting Lightning Charger for 2016 Macbook Pro to 2021 Macbook Air M2

2016 MacBook Pros had USB-C (Thunderbolt 3) ports, and charged through them.

Your M2 MacBook Air has both USB-C (Thunderbolt 3) ports and a MagSafe 3 port and can charge through any of them. So if your display connected to your 2016 MacBook Pro with a USB-C cable or a Thunderbolt 3 cable which carried both video and power, it should be able to connect to your M2 MacBook Air in the same way.

Charge your MacBook Air or MacBook Pro - Apple Support

I'm assuming that your "display with a Thunderbolt charger" is not an Apple Thunderbolt Display (27-inch) (model number A1407). That display had a separate charging cable with a MagSafe 1 connector that would not have been compatible with your 2016 MacBook Pro – and that is not compatible with your M2 MacBook Air's MagSafe 3 port. Either notebook could send video to that display with the help of an Apple Thunderbolt 3-to-2 adapter, but both of the notebooks would need to draw their power from some source other than that display.

If that is an Apple Thunderbolt Display (model A1407), I believe the power cable would have a MagSafe 1 plug.

There is (or was) an adapter to go from MagSafe 1 to MagSafe 2 – but as far as I know, there are no adapters to convert either of these to MagSafe 3 (the type of MagSafe connection on the M2 MacBook Air).

You'll probably need to use the charger that came with the M2 MacBook Air to power/charge the computer.