If I first bought the game on Steam is it possible to unlock the full version here? Buying it again to be able to try it on phone would be kinda difficult, and I think Itch.io has a link option with steam accounts

no they would usually state on itch io that if you buy the game from here you would get a steam key ect ect, but it doesnt state that plus i dont think it would be fair to get it free just because it was paid for on steam