Oh I should of mentioned this.
Traveling from Taipei to Tainan is like taking a bus. Nobody ever calls ahead of time to book the flight. You go to the airport and buy the tickets on the spot. So getting window seats is hard. FAT offers flight to Tainan every single hour from 6 am to 10 pm i think. And so do all the other carriers there. And I do not think they are part of any alliance, however I think UniAir is Eva Air's domestic version, seeing as how the logo is almost the same, infact I even thought it was Eva Air until I saw the Uni part of the name.
Thanks for reading my report!