Straight up, I’ve got to say that I’m not certain about the identity of “the angel of the Lord.” The Bible isn’t definite about it. However, there are some indicators as to who the angel is. I’ll list three of them, in which the definite article “the” is used to differentiate the angel from all other angels. The indicators are in the words spoken by the angel.
1. The angel of the Lord said to her, “I will greatly multiply your descendants so that they will be too many to count.” Genesis 16: 10.
The angel is speaking as God, who spoke similar words to Abraham in Genesis 22: 17 and 26: 4.
2. But the angel of the Lord called to him from heaven and said, “Abraham, Abraham!” And he said, “Here I am.” He said, “Do not stretch out your hand against the lad, and do nothing to him; for now I know that you fear God, since you have not withheld your son, your only son, from Me.” … Then the angel of the Lord called to Abraham a second time from heaven, and said, “By Myself I have sworn.” Genesis 22: 11, 12, 15, 16
It can be construed that the angel is identifying itself with or as God. Also, the phrase “By Myself I have sworn” is identified as being uttered by God Himself in Isaiah 45: 23 and Jeremiah 49: 13.
3. Now the angel of the Lord came up from Gilgal to Bochim. And he said, “I brought you up out of Egypt and led you into the land which I have sworn to your fathers.” Judges 2: 1
In the preamble to the ten Commandments, God reminds the Israelites that it is He who freed them from Egyptian bondage. (Exodus 20: 1).
These indicators seem to suggest that the angel of the Lord could be a manifestation of God. In some instances, the manifestation would have been in human form. That was the case in Judges 13: 1-22. There we read that the angel of the Lord appeared as a man of God.
Bible scholars have long maintained that God showed up in human form before Christ’s incarnation. Another instance is in Genesis 18, whereupon Abraham met and served food and drink to three men. Abraham recognized one of them as the Lord.
Scholars have a couple of names for the pre-incarnate, physical appearances (including the burning bush) of God. If done by the Father, they call it a theophany. If done by Christ, it’s called a christophany. Incidentally, “the angel of the lord” is never used in the Bible after Christ’s incarnation.
In any case, even if in some scriptural passages “the angel of the Lord” can sometimes be inferred as simply a unique angelic spokesman, the term overwhelmingly seems to me to be more likely referring to God Himself.