I shoot RAW with my Nikon D700. All of my NEF files show "Display P3" as the color profile in File > GetInfo. I import (actually, transfer) the NEF images from my CF cards directly to my Mac using a card reader, so there is no intermediate software 'working' on the files when I import them, they are direct copies from the CF card; so I cannot say where or how the "Display P3" gets associated with the files, although I suspect it may be a default placeholder and may only be shown in GetInfo without actually being associated with the image.
I tested a number of my NEFs this evening and did not see any oversaturation or any other visual effect that would suggest an out-of-order color profile. I checked them in the Finder, Preview, Adobe Camera Raw and Capture One. In all cases the images presented 'naturally' - meaning everything as expected, no apparent changes, nothing odd.
When I checked the metadata on these same images (in Photoshop) I did not see anything indicating that any color profile was associated with or embedded in any of the images.
My understanding of when a color profile is actually embedded in a RAW file (normally via an in-camera setting, either Adobe RGB or sRGB, never Display P3) is that it has no effect on the RAW image itself, that it only affects the JPEG preview of the RAW image. In ACR you can set or change the color profile associated with a RAW file but those are all Adobe profiles ... Display P3 is not one of the choices.
Photos does display JPEG previews of RAW images by default unless you set Image > Use RAW as Original.
What model Mac are you using, and what Display Profile is it set to use? (See System Preferences > Displays > Color)
I hope some of this information is helpful or at least suggests some avenues to explore. I would not rule out anything as to why your images look the way they do ... first thing I would check is all in-camera settings. I'd also import a few images into yet another computer (perhaps not a Mac, or at least one not on Monterey) to see if the effects show there also. If so, it could be a camera problem.