iMAC 2019,1 (macOS Catalina) as secondary monitor for MacBook Pro 2021

Not with Target Display Mode. It is dead tech and both of your devices are too new for TDM anyway.

The remaining options for reusing the iMac as primary display for the Macbook Pro are all less than satisfactory. You iMac supports AirPlay to Mac but most videos I see posted of it in action show it as quite laggy. Network screen sharing is also laggy.

Third-party option are expensive can can be laggy. One, Luna Display, has this disclaimer on its main support page:

If you need an external display for the Macbook Pro, save yourself a lot of grief and buy a regular external monitor:

• No lag
• easier to set up
• only a bit more expensive than the third-party reuse options
• more flexible display options
• uses far less power than a fully running iMac.

+1 Allan is spot on when he says, "save yourself a lot of grief and buy a regular external monitor"

see > Use an external display with your MacBook Pro - Apple Support