Interesting reasoning to think about in these statements:

How could "God" be an "Angel" of himself?

THEODORET OF CYRUS (circa. 393-457 C.E.): "...The whole passage (Exodus 3) shows that it was God who appeared to him. BUT (MOSES) CALLED HIM[*] AN ANGEL in order to let us know that it was not God the Father whom he saw — FOR WHOSE ANGEL COULD THE FATHER BE?BUT THE ONLY-BEGOTTEN SON, THE ANGEL OF GREAT COUNSEL..." - (Commentary "In Exodus" Roberts & Donaldson ANF.)
[FOOTNOTE]: Refering to Jesus as the Angel of the Lord in the OT.

Compare the above with a statement from earlier Christians:

COUNCIL OF ANTIOCH (circa 264-269 C.E.): "...Sometimes he is testified to as "an Angel," but sometimes as "a lord," but also sometimes as "a god." FOR IT IS TOO UN-REVERENTIAL TO EVEN THINK OF CALLING THE DEFINITIVE GOD OF THE ENTIRE UNIVERSE “AN ANGEL.” But the Son is thee Angel of the Father. Being himself a lord and a god. As it is written: “[the] Angel of His Great Purpose[*]..." - (Epist. Synodic. ad Paul. Samos. Translated by Matt13weedhacker 28/08/2011)
[FOOTNOTE *]: Isa 9:6; LXX., or "...Angel of Great Counsel..."

HENRY NEWMAN: "...Again, the Council of Antioch, which condemned Paul of Samosata, says that He "...appears to the Patriarchs and converses with them, being testified sometimes to be an Angel, at other times Lord, at others God;" that, WHILE "IT IS IMPIOUS TO THINK THAT THE GOD OF ALL IS CALLED AN ANGEL, THE SON IS THE ANGEL OF THE FATHER[15]..." - (Section 2. Our Lord's Incarnation and the Dignity of His Blessed Mother and of All Saints, Chapter 4. Instances in Illustration, NEWMAN READER.)
[FOOTNOTE 15]: Reliqu. Sacr. t. ii. p. 469, 470.

Even as late as the Council of Chanceldon the tradition of calling Christ an Angel continued in some form or degree:

Encouraging faith in the very words of Isiah 9:6 LXX when writing against Nestorious:

COUNCIL OF CHALCEDON (A.D. 451): “ with faith the words of the same prophet, Unto us a Child has been born, unto us a Son has been given, whose power is on his shoulder; and they shall call his name Angel of great counsel, Wonderful, Counsellor, Strong God, Prince of Peace, Father of the age to come...” - (Extracts from the Acts: Session II Continued. Translated by Henry Percival. From Nicene and Post-Nicene Fathers, Second Series, Vol. 14. Edited by Philip Schaff and Henry Wace. (Buffalo, NY: Christian Literature Publishing Co., 1900.)