Your monitor’s screen resolution will impact the available settings and external screen capabilities.
While undocumented, testing has demonstrated that 4:3 monitors with a display resolution of 1280x1024 and below are currently only usable with iPad screen mirroring; as such, associated settings to disable screen mirroring in favour of Stage Manager are not available in iPad Display settings.
When used with a 16:9 widescreen monitor, with a native display resolution of 1920x1080 or greater, the full set of Display settings - and access to Stage Manager - become available.
The iPad User Guide lists the upper resolution limit as 6K - but the Guide offers no lower limit, or suggestion of aspect-ratio, being a factor for compatibility:
Connect iPad to a display with a cable - Apple Support
This discussion thread may be of additional interest:
iPad Extended Desktop not working - Apple Community
In conclusion, your monitor must be capable of 16:9 widescreen 1920x1080 (FHD) to support iPadOS16 Stage Manager extended desktop. Unless the monitor meets the minimum required specifications, you will not see the full range of controls in iPad settings.
