That‘s at least the default behavior if not the only way to play video on iPhone‘s Safari. iPad Safari however supports non-fullscreen video playback.
Fullscreen is definitely not the only way to play video on iOS safari...
iOS Safari can play videos non-fullscreen with the playsinline or webkit-playsinline attributes. The fun thing though is that this attribute must be enabled manually on each web view, so if some other app is embedding your page and they didn't enable it, your videos will play fullscreen despite the attribute being set.
YouTube (the website) plays video in a frame just fine.
The “play” method is enough to trigger the full screen — which, by the way, isn’t really intuitive imho.
That's only if the video doesn't have the attribute `playsinline`.