Not a mobile dev so I'm leaning quite heavily on other people's experiences here.
It's my understnading that setting up cookies to work on mobile is quite painful, though it depends on the platform. IIRC iOS has gotten better at it with a shared cookie storage, but Android requests are still stateless by default, so you basically have to manually wire up a cookie jar and carry it around everywhere you go, so to speak.
Attributes like SameSite also behave differently, and WebViews don't share the same cookie jar as native requests as far as I understand.
Bottom line is that maybe "Cookies also don't work on mobile" is a bit of a wrong statement, but it's certainly more of a hassle and a path lined with more footguns than just wiring up an OAuth provider and sending access and refresh tokens back and forth using Authorization headers. The great thing with Session cookies on desktop is simplicity, which you sort of lack on mobile.