I wonder how the whole thing works when I open a youtube video from a preview inside a chat application on my mobile phone.

It looks like the video loads and starts playing in some kind of in-app browser, but there is just full-screen video and nothing else. I also never faced any ads in this "mode" of playing a video, yet recently some strange things started happening where the playback would start together with an audio-track from the advertisement. The video itself would start playing but the sound would be replaced with the sound from ad which seemed very odd and much like a bug, only when advertisement audio track ends it will be replaced with audio track from the video itself.

I'm genuinely curious how is the whole playback process different when I watch a video from the Telegram preview, can I somehow achieve the same "just fullscreen video" kind of playback on the desktop as well? Does anyone have any insight?

Not sure what you’re asking.

You can get fullscreen video in a desktop browser by pressing F or clicking the fullscreen icon (broken rectangle) below the video.

You can get ad-free playback by paying for Premium or sometimes with an ad blocker.

Outside the browser, you can get both with yt-dlp, which also integrates into video players like mpv.