iPad Air 5th Gen M1 only mirroring on external display, not extending the display.

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.  

When mirroring your iPad display to an external monitor, expect to see black bars to infill where the iPad screen doesn’t match the aspect ratio of monitor screen.

Whether or not this is an intentional limitation, or a bug introduced with release of iPadOS 16.4.1 or later, is unknown.

Activate stage manager.

Then go to "settings" menu.

Choose "multitasking & …".

In the following menu choose "stage manager" for your ipad and as well "stage manager" for your external display (note: menu for external display only shows up as long as you have connected the external display).

Now the external display should show up as extended display.

As illustrated, your “Dock Station” appear to be interfering with detection of the external monitors capabilities - perhaps as the dock itself is attempting to strip audio and [possibly] output to a different port.

On the basis of available information, I would speculate that your dock is not well suited for use (or necessarily fully compatible) with your iPad.

Usually, if speakers are detected, iPad will unconditionally direct audio to the monitor in favour its built-in speaker system. That said, there are recorded instances where iPad will maintain audio with the iPad’s speaker array.

Depending upon you particular situation, you may be able to re-direct audio from the iPad Control Centre. Swipe-down from the upper-right corner of the screen to expose the Control Centre:

Touch-and-hold (i.e., a long-touch) the Now Playing box (above):

Now tap the button identified above:

You may find the monitor listed here.

You are very welcome. If you need any additional help, don’t hesitate to return to this thread.