The phrase that is usually translated “the Angel of the Lord” shows up in in the Old Testament 67 times (including the description of the character in the burning bush). It is a curious phrase considering what it leads us to ask about the identify of this angel and its relationship to God. The Hebrew here is “mal’ak yahweh”, and because of how its constructed must be a specific definition (THE angel of the Lord) as opposed to something more broad (AN angel of the Lord). Think similarly of how we might see the phrase “the River Euphrates”. We recognize that this is a specific river called the Euphrates, not a river in or of Euphrates. The second word is used to provide clarity on how to understand the specificity or use of the first one. (This is coming to a point, I promise).
So, given those rules, it’s possible we could read the phrase “mal’ak yahweh” as “the angel that is Yahweh” or “the Angel Yahweh”. But this seems problematic. We know that God is not an angel, per se, or is at least not limited to that distinction in a moment in time. The other option is some kind of impostor angel who takes on God’s identity in certain situations.
Complicating this further is the number of Old Testament passages where “mal’ak yahweh” is either called or recognized as the Lord. Remember in Genesis 16, we have an Angel of the Lord (v.11) who talks to Hagar and promises she will be safe. In response, it says, “…she called on the name of the Lord who spoke to her, “You are a God of seeing.” Who spoke to her? The Angel of the Lord…who she just called God (or at least attributed God qualities to.) Or Genesis 31, when the Angel of God shows up to Jacob to tell him to leave Laban and go home. In v.13, the angel says about itself, “I am the God of Bethel…”
The strongest of all these connections happens in Exodus 3, within that burning bush. In v. 2, it is the Angel of the Lord who first appears in the burning bush. When Moses turns to see, in v. 4 it says that the “Lord” saw and that “God” called out to him from the bush. Unless there’s a switcheroo happening that is omitted from the text, it sure seems like the Angel of the Lord is indeed God Himself.
That leaves a reasonable question as to why. Why not show up as “God” instead of an Angel of the Lord? To a certain extent, trying to nail down the motivations of God and, the human rendering of their interaction with God, will often feel unsatisfactory. Often, we can’t really know, we can just speculate. But I have two thoughts. One, the identification seems to progress. We start with a description of a messenger/angel/being who is acting on the Lord’s behalf. And then as the narrative progresses the true identity is realized. Further, given how we understand the nature of God, specifically his omnipresence, it’s difficult to conceive that He can be represented at a specific moment/location in time, especially embodied in something that appears human (not necessarily the bush here, think about the Hagar interaction, wrestling with Jacob, etc.)
So how to think about it? It think it’s pretty this is the presence of God vs. some sort of other being acting on His behalf. In resolution of the representation a specific moment/location in time vs. omnipresence, think of it like a video-conference. You get the sense of the presence of someone in a room through a video screen and speaker, even though they are not fully present.
Alternatively, this is where we find the thought that this is a pre-birth (or pre-incarnate) Jesus. We have experience in Jesus understanding a God who can be physically manifested in tandem with retaining all the attributes and authority of God (being still God, of course). Thus, when we see the phrase “the Angel of the Lord”, we can think of it as God, but specifically Jesus.
It’s not a slam dunk for me that we are to see that as always specifically Jesus. But given that we should rightly see the Angel of the Lord as God Himself in some form or another, I’m not sure it particularly matters.