Most software products other than Canon DPP do not recognize settings set in the camera (I hesitate to say all because there may be some out there that do, but I have never seen any identified).
Recentlly, with the release of Lightroom Classic 9.2, it is now possible to have Lightroom acknowledge and use a Picture Style that has been set in camera.
One caveat is that there needs to be a camera specific profile option in LR for the camera. There are currently no LR camera specific presets for cameras using the .CR3 format.
You can verify correct setup by shooting a RAW image with the Monochrome Picture Style selected. If you have set everything up correctly the image will stay monochrome when imported and rendered in LR.
09/26/2023: New firmware updates are available.
08/18/2023: Canon EOS R5 C training series is released.
07/31/2023: New firmware updates are available.
05/31/2023: New firmware updates are available.
05/18/2023: New firmware updates are available.
03/30/2023: New firmware updates are available.