Akagami no Shirayuki-hime, Saijaku Muhai no Bahamut Recommendations
Both animes have a prince and a princess like figure in them. Both animes also use swords to fight their battles, but unlike Snow White with the red hair saijaku no bahamut also use robot mechs to fight with alongside with a sword. Both animes have similar main characters in terms of them having a sense of protection towards others (main girl protagonist). Each of the animes are set in a kingdom for saijaku no bahamut tis would be "the new kingdom" and for Snow White with the red hair this would be "Clarines".