Targus Doc With Display

The M2 MacBook Air only supports one external display.

Macs do not support DisplayPort MST daisy-chaining or the equivalent. As a rule, Macs that can drive multiple external displays over USB-C or Thunderbolt support connecting

• One display to a USB-C dock
• Up to two displays to a Thunderbolt dock

Targus – USB-C Universal DV4K Docking Station with 100W Power Delivery

This does not look like a Thunderbolt dock to me. The description says "USB-A and USB-C, DP Alt. Mode, TB3 Compatible," but lots of vendors who make peripherals that depend on USB and DisplayPort Alt Mode will make claims that their devices are Thunderbolt-compatible.

So

• Your Mac only supports one external display.
• This monitor supports two external displays in some way (DIsplayPort MST daisy-chaining, or a "workaround" technology like DisplayLink) that would not work, without drivers, even on a more capable Mac.
• If you review the dock manual, you will very likely find that you are supposed to install a special driver on your Mac. Workarounds depend on the combination of a "magic decoder ring" chip set in an external adapter/hub, and a matching driver on the host computer. Neither the device alone nor the driver alone are enough.
• If you can get things to work, you may find that the arrangement comes with some compromises.

In addition to its integral display your MacBook Air M2 supports one external display.

Use an external display with your MacBook Air - Apple Support

Check with the dock manufacturer to determine compatibility.