The identical actions of the Lord and the Angel of the Lord show they are the same Being. The God known by Abraham, Isaac, Jacob, and Moses was Jesus Christ.
Christ frequently used 3rd person titles, such as the Son of Man and the Helper. Just as Christ sent the Helper—Himself—so Yahweh sent His Angel—Himself.
The Jewish converts to the Way, although having had the benefit of Messianic prophecies, did not recognize the powerful significance of Psalm 8.