The origin of "Angel of Music" as pertaining to the Phantom are back in Christine's childhood. The references are all to the original Gaston Leroux's novel.
In Chapter V "The Enchanted Violin", we first learn that Christine have beend told the tales of the The Angel of Music from her father:
"Little Lotte thought of everything and nothing. Her hair was golden as the sun's rays and her soul as clear and blue as her eyes. She wheedled her mother, was kind to her doll, took great care of her frock and her little red shoes and her fiddle, but most of all loved, when she went to sleep, to hear the Angel of Music."
The Angel of Music played a part in all Daddy Daae's tales; and he maintained that every great musician, every great artist received a visit from the Angel at least once in his life.
No one ever sees the Angel; but he is heard by those who are meant to hear him. He often comes when they least expect him, when they are sad and disheartened. Then their ears suddenly perceive celestial harmonies, a divine voice, which they remember all their lives. Persons who are visited by the Angel quiver with a thrill unknown to the rest of mankind. And they can not touch an instrument, or open their mouths to sing, without producing sounds that put all other human sounds to shame. Then people who do not know that the Angel has visited those persons say that they have genius.
More importantly, her father told her:
"You will hear him one day, my child! When I am in Heaven, I will send him to you!"
This made an impression on her, and when Erik started teaching her, she assumed he was the Angel her father promised her before he died:
"Listen, Raoul. I have decided to tell you something serious, very serious ... Do you remember the legend of the Angel of Music?"
"I do indeed," he said. "I believe it was here that your father first told it to us."
"And it was here that he said, 'When I am in Heaven, my child, I will send him to you.' Well, Raoul, my father is in Heaven, and I have been visited by the Angel of Music."
Now, as to why he calls Christine "Angel" in the musical, there are two reasons:
First, he sees Christine as angelic, from the source book (Chapter XXVI "The End of the Ghost's Love Story"):
You're crying, too, daroga ... and she cried also ... the angel cried! ..." Erik sobbed aloud and the Persian himself could not retain his tears in the presence of that masked man, who, with his shoulders shaking and his hands clutched at his chest, was moaning with pain and love by turns.
Additionally, she inspired his music, just like the "Angel of Music" of the tales - which is probably the significance you seek.