PS2 emulation on the Xbox has always left a lot to be desired. Texture issues, missing skyboxes, and framerate problems plagued its games, but a lot of that has been remedied with the XBSX2 app as seen in Modern Vintage Gamer's newest video.
An older emulator, RetroArch, used the PCSX2 core, bringing PS2 games to Xbox for the first time, but it had a lot of those aforementioned teething issues. They've since been ironed out with a standalone app from Stenzek, and this standalone app even has internal resolution scaling, letting you play games in native PS2 resolution, 1080p, and even 4k. However, the latter isn't recommended for Series S.
This app wasn't made publicly available, so the community put together XBSX2, "A fork of PCSX2 built specifically for the Xbox." In MVG's video, we can see Ratchet & Clank running smoothly and upscaled, as well as God of War and Burnout 3. To use the emulator, you'll need a PS2 bios which you can get if you own a console, as well as the disc files for individual games.
This means that, if you have the right tools, you can play everything from classic Xbox to classic PlayStation titles in one place, running at higher resolutions and framerates than they originally did. And if you run into any issues with rendering or textures while using XBSX2, you can switch to software rendering on the fly and resolve these problems. This will lower resolution and go back to that fuzzier retro PS2 look, but you'll get an authentic and, importantly, less buggy experience.
Older emulators also couldn't handle playing games natively at 60fps, whereas this new emulator is more than capable. Choppier audio and visuals are now resolved, even at double resolution.
It's important to note all of this is community-made. You won't find Xbox releasing an official emulator for its competitor, though Valve did accidentally feature a Switch emulator in a new Steam Deck video. So if you want to get it up and running, you'll have to fiddle with downloads and apps yourself.