Before the launch of iPadOS 16 (iPadOS 16.1), Apple removed a feature from Stage Manager that allowed Stage Manager to be used with external displays. In iPadOS 16.2, all of this changes - but there is a catch.
The removal of the external display function was done to extend the Stage Manager support to the iPad Pro models A12Z and A12X. With the new beta version of iPadOS 16.2, the developers on Tuesday evening receive Apple has re-enabled support for external displays on iPads with an M1 or M2 chip.

iPadOS 16.2 expands the functionality of the Stage Manager
While the Stage Manager works on older iPad Pro models without an Apple Silicon chip, these iPads cannot use the Stage Manager with an external display. By supporting the external display, up to eight apps can be used with the Stage Manager instead of the four available on the iPad alone. (Image: Apple)