Nope, no special API what so ever. In fact there's even a Firefox add-on that makes it work[0], Google Meet just doesn't target more than Chromium with this feature.
If you try using it, they'll redirect you here[1], and tell you to:
> Check if your browser supports WebGL 2.0 at webglreport.com and verify that “Major Performance Caveat” is marked as “No”.
...but clicking on that site from Firefox shows WebGL 2.0 support and "Major Performance Cavet" is indeed marked as "no". That being said, simply changing the user agent to Chrome also doesn't work.
So in conclusion, I end up using Chromium a few hours per week purely for Google Meet's blurry backgrounds, and I use Firefox for everything else.
Played with the extension, it does not seem to enable this feature: the option appears, but then there's an error message. Although the other features are fun.
It works for me on Chrome but I had to use Wayland (not Xorg) and then force enable GPU rendering because it's currently blacklisted for Wayland. Possibly relatedly to that blacklist every couple of days (even with rebotos) new tabs in an existing browser window stop rendering content and I have to quit and reload. That doesn't seem to happen on Xorg/without GPU.
Kinda. Google uses the WebAssembly SIMD extension for their segmentation. Not sure if they have a fallback on Firefox, but if they do, it most likely won't perform the same way.