He has no other name.
From his biography,
Shan Yu is the main villain from the Disney animated movie "Mulan". He is the leader of the Huns.
He appears in "Kingdom Hearts II". There, he tries to use the Heartless to take over China, but Mulan, Sora, Goofy and Donald defeat him.
There are many other characters without a specific name in Mulan, such as The Emperor and The Matchmaker. Maybe Disney didn't think the eastern names were that easy to memorize by western audiences? Another option, as R.M. puts it, "they thought they didn't need names. For example, the Evil Queen in Snow White is never named in the movie, nor is the Fairy Godmother in Cinderella. -- A fair number of even somewhat significant characters in Disney films aren't given actual names."
Anyway, if Shan Yu is meant to represent the leader of the Huns, Disney might have been referencing Attila, the Hun (who in fact invaded Europe, not China), or some undisclosed leader of the Xiongnu (the alleged predecessors of the Huns who did invade China). Another source of inspiration, according to this, is Genghis Khan, leader of the Mongol armies (who also invaded China).
His falcon is known as Hayabusa.
Hayabusa is Shan Yu's pet falcon and sidekick. The bird often acts as a warning signal, alerting China's victims that Shan Yu is near. He appears to be similar to his master: strict, ruthless and vicious. He latches onto his prey or whatever he needs to carry with his steely talons (as he did with Shan Yu's sword when Shang handed it to the Emperor).