Well..look at mine.. it is original. Mine flange is metal and a bit rusty..but it still functional. Someone went to HD and got a flange and made new holes in your shower to make something work with the to toilet. Maybe 30 years ago.. But they make what you need at HD Lowe's or Menards ..
YES it screws in.. But I dont know the thread size, i tossed my tank, building a new one
You need this one as it is 3in.
test fit then cut off any material to make it the size you need to match