The Apple Studio display has a full version of the software.

In System Information, under Graphics/Displays, you can see the same build number as iPadOS and iOS.

The A13 Bionic chip is the same chip as the one used in the iPhone 11 lineup and supports a 12-megapixel Ultra Wide front-facing camera with Center Stage and six-speaker sound system. While the presence of the A13 chip indicated that the display likely ran a variant of existing Apple software, in much the same way that the HomePod and HomePod mini run a version of tvOS, the device's exact software was unknown until now.

studio display firmwareThe Studio Display's firmware in System Information (via Daring Fireball).

Yesterday, the first reviews of the Studio Display were released, with a common complaint among reviewers being that the built-inWebcam's image quality ranges anywhere from an oldBlackBerry to downright awful in their hands-on testing. In his detailed review of the studio display, he said that the camera is terrible, the image is terrible, and the center stage isglitchy.

Even without harsh sunlight, all images from the Studio Display camera, in all lighting conditions, are grainy, lacking in contrast, and make skin tones look cadaveric.

[...]

... How can the image quality from the camera on a $1600 display be so much worse — laughably worse — than the image quality from a $600 iPad Air that uses the exact same camera hardware? Let alone comparing it to the front-facing camera on the $430 iPhone SE, which makes the Studio Display camera look like a toy. And we waited years for Apple to ship this display. Again, it's usable. All sorts of people use way worse cameras for videoconferencing every day. But this image quality is embarrassing from a company that considers itself the leading camera company in the world... I expected to be impressed by the Studio Display camera. Instead, I'm baffled. I don't understand how this shipped.

It gets even worse. The Center Stage feature on the Studio Display should be called Off-Center Stage. Move around a bit or turn your head to the side and you get framed off to the side, even though you're sitting directly in front of the center of the display. It takes up to 5 seconds for Center Stage to catch up and re-center you in the frame, which it does slowly and sheepishly, as though it's embarrassed...

Apple told reviewers that it discovered an issue where the system is not behaving as expected and will be making improvements in a software update. The author concluded:

The Off-Center Stage thing is obviously a bug, and I expect that to be fixed. The overall image quality, I'll bet, can and will be improved to some degree via software updates, but I'll be surprised — happily surprised, but surprised — if a software update can turn this camera into something Apple should be proud of. Maybe, though, given that it's the same camera hardware as the front-facing camera on the new iPad Air and last year's iPad Pros. But I'm not holding my breath.

With the confirmation that the studio display runs the operating system, Apple will likely make an update to improve the quality of the webcam as part of an update. There is no information about how software updates for the studio display work are made.