The description for Flying Press in SwSh says "Is it flying and fighting type simultaneously." Would the Pokemon turn into a Flying/Fighting type or just Fighting becuase it is labeled as Fighting?
It's base type is fighting. So you'd just become a plain fighting type. It just happens to combine flying in it's type effectiveness, but it's not an actual flying type move, it's fighting. Here's a showdown replay to show what happens.
I hope this helps!
It turns into a fighting move.