what's in a name.
Simca - Migratory Bird (to her face).
Sora - Takeuchi Sora.
Ikki - Wind King or Sky King, depending on the context and the string of words he uses. Itsuki Minami has been used, yes.
Ringo - Thorn Queen to others. I'm sure he calls her Ringo in private conversation. He's not that much of a condescending asshole.
Kururu Sumeragi - Pledge Queen.
Kaito - Shinjuku Crocodile, Wanjima Kaito.
Rika (NOT THAT SHE'S HERE BUT WOW SHOULD SOMEONE APP HER) - Rika.
Ine - Makigami Ine, former Pledge Queen,
Spitfire - Spitfire, Flame King, former Flame King?
Akito/Agito - The Fang King.
A lot of the above are grabbed from canon. Most of Ikki's team is brushed off in canon as sucking, so that's that. He refers to most people by their full name and will likely do that with anyone he meets. Any remaining Kings will be called by their titles.
Anyone he calls by a really formal name, with the EXCEPTION OF KAITO, was probably called something a lot nicer when he was younger. Simca was Simca, Spitfire was Spitfire, there wasn't any of that Kings crap thrown in. He refers to titles because it shows a mixture of disdain and respect for the remaining AT world. Chances are, he has data, or had data, on most people heavily involved in the AT world. Emiri or Yayoi would probably be excluded from this.