A lot of the Destroyers in Kancolle tend to call the player "Shikikan" while i notice a lot of the Battleships call the player "Teitoku".
The term "Shikikan" isn't used in Kancolle at all (unless one of the newer ships uses it (I kind of haven't played in a year)); they use "Shireikan" if at all.
A lot of the Destroyers in Kancolle tend to call the player "Shikikan"
It is "Shireikan".
while I notice a lot of the Battleships call the player "Teitoku".
Off the top of my head. Inazuma sisters vs Kongou sisters.
It seems like shipgirls who call the player admiral were commanded by the real admirals in real life. Maybe that's why most Kancolle battleships use the word "Teitoku".
Some battleships in Kancolle, such as Hiei or Kirishima, still use the word "Shirei".