In ancient Middle Eastern culture, authorized servants were often empowered to act on behalf of and in the name of their masters. One example of this is Abraham sending his servant to find a wife for Isaac (Genesis 24:1-9). Significantly, although scholars believe the servant was Eliezer of Damascus, he is not named.1 When he goes to negotiate on behalf of Abraham, it is as if he were Abraham.
This divinepractice of investing of one's name/authority in an authorized representative is also present in Jesus' teachings in the New Testament (Matthew 10:40; John 5:43). In this context, we could refer to it as a "divine investing" of the authority of God's name. This is what the angel would be doing in the context of the passage you mention.
1 Kenneth L. Barker, ed., NIV Study Bible, Fully Revised Edition. (Grand Rapids, MI: Zondervan, 2020), 51.