Making the browser itself pretend there's a "desktop" camera would be technically possible, but since it would only work with that single browser, it doesn't look like anyone has done this yet. (At least I couldn't find any work for Chrome or Firefox – even though they have 'screen sharing' as a WebRTC feature, nobody seems to have linked it to webcam API yet.)
However, there are products which install a custom device driver that creates a virtual "webcam" device, which is then usable by any program. Your web browser will think you have two webcams; you still need to approve camera requests, and it's still up to you to choose the "preferred" camera.
I googled "virtual camera device" and found several products which can stream the desktop, for example Webcamoid (appears to be multi-platform and open-source), VCam (commercial), OBS VirtualCam (a plugin for OBS).
(For Linux there are two 'generic' drivers, akvcam or the older v4l2loopback, but they still need a program that would provide the actual video data. For Windows and macOS each product most likely uses its own custom drivers.)