screen mirroring between a 2023 Mac Air and a 2019 iMac 5k display

Hi,

I think you can't use Target Display Mode on your iMac. If you want to mirroring display, try to Screen Sharing which is described following document.

Share the screen of another Mac – Apple Support (AU)

Target Display Mode had been dead for many years, your Macs are too new to support it. The fastest, most reliable and likely least expensive solution is a simple third party external display.