@Talh
yes I fully aware about that, but if we're talking about this series then Byakko is her epithet, Byakutei is her title, and Kohaku is her name. You even can see the kanji for Kohaku (琥珀) that Touya wrote on the ground.
Also there is no peculiarity about "white tiger" either, that's basically just how Japanese people say white tiger in Japanese and at the same time, the mythical beast is a tiger with white fur. So if an English speaking people addresses this beast, it will be "the white tiger".