However Jesus was talking about someone specific in relation to Abraham during this entire discussion, so he was referring to being that person. Just as the blind man also said the exact same phrase I AM in relation to what he was being asked in John 9:9 just a few verses later.
Jews were looking for the Messiah who was sworn to be like the Melchizedek who served Abram before he was named Abraham.
Psalms 110:4 The LORD hath sworn, and will not repent, Thou art a priest for ever after the order of Melchizedek.
So you will find commentary and explanation of John 8 in Hebrews.
John 8:53 Art thou greater than our father Abraham, which is dead? and the prophets are dead: whom makest thou thyself?
Hebrews 7:4 Now consider how great this man was, unto whom even the patriarch Abraham gave the tenth of the spoils.
John 8:54 Jesus answered, If I honour myself, my honour is nothing: it is my Father that honoureth me; of whom ye say, that he is your God:
Hebrews 5:4-6 And no man taketh this honour unto himself, but he that is called of God, as was Aaron. So also Christ glorified not himself to be made an high priest; but he that said unto him, Thou art my Son, to day have I begotten thee. As he saith also in another place, Thou art a priest for ever after the order of Melchisedec.
John 8:55-56 Yet ye have not known him; but I know him: and if I should say, I know him not, I shall be a liar like unto you: but I know him, and keep his saying. Your father Abraham rejoiced to see my day: and he saw it, and was glad.
Hebrews 7:1-2 For this Melchisedec, king of Salem, priest of the most high God, who met Abraham returning from the slaughter of the kings, and blessed him; To whom also Abraham gave a tenth part of all; first being by interpretation King of righteousness, and after that also King of Salem, which is, King of peace;
John 8:57-58 Then said the Jews unto him, Thou art not yet fifty years old, and hast thou seen Abraham? Jesus said unto them, Verily, verily, I say unto you, Before Abraham was, I am.
I AM THE MESSIAH WHO IS LIKE MELCHIZEDEK
So all this has nothing to do with Jesus saying he is the God of Moses, but he is saying he is this human Messiah who is a priest like Melchizedek as promised in Psalms.