The context indicates that there is no correlation or contrast - egó and egó eimi merely being the first person singular.
Me, myself, I.
The translation in Mark 12:26 of egó or "I" into "I am" is not precluded by the context. However, it is not strictly necessary or indicated. Some translations note this.
I [am] the God of Abraham, and God of Isaac, and God of Jacob ... (Mark 12:26 Rotherham)
Various forms of the AKJV have "am" italicized for this reason.
In Mark - egó - and the parallel account in Matthew (22:31-32) - egó eimi - Jesus references a declaration made by "God", (at Exodus 3:6 and echoed at verse 15), establishing his authority regarding a commission given to Moses. This context made clear through Jesus prefatory interrogative :
... Have ye not read in the book of Moses, at the Bush, how God spake unto him, saying––I [am] ... (Mark 12:26 Rotherham)
From the account in Moses :
And he said––I, am the God of thy father, God of Abraham, God of Isaac, and God of Jacob. (Exodus 3:6 Rotherham)
An analog found elsewhere in Moses :
And he said, The sound of thee, heard I in the garden,––and I was afraid, for, naked, was I, so I hid myself. (Genesis 3:10 Rotherham)
In both instances, וַיֹּ֕אמֶר being "he said" but more pertinently, "I" from אָנֹ֖כִי - the first person singular.
Me, myself, I.
It is not until later, when the declaration is repeated, that the divine name is included in the declaration :
And God said yet further unto Moses––Thus, shalt thou say unto the sons of Israel, Yahweh, God of your fathers, God of Abraham, God of Isaac, and God of Jacob, hath sent me unto you. This, is my name to times age–abiding, And, this, my memorial to generation after generation. (Exodus 3:15 Rotherham)
This of course resulting from the divine name being declared at the previous verse :
And God said unto Moses, I Will Become whatsoever I please. And he said––Thus, shalt thou say to the sons of Israel, I Will Become hath sent me unto you. (Exodus 3:14 Rotherham)
At the verse Jesus quotes, the divine name had not yet been declared.
This is understood in Mark and Matthew by the use of egó and egó eimi respectively. This is the Greek rendering of the אָנֹ֖כִי of Exodus verse 6 - the first person singular pronoun.
The Greek text for verses 6 and 15 being :
egō eimi ho theos ho patros ... (verse 6 LXX)
kyrios ho theos ho paterōn ... (verse 15 LXX)
If Jesus had intended to refer to the divine name, rather than stating egó or egó eimi - simply "I [am]", we would be seeing κύριος (kyrios) or "lord".
Regardless, the context is clear, from the prefatory interrogative.
Jesus is quoting a statement - made some 1500 years before his birth, by his "God", to the man Moses.
... have you not read in the Book of Moses, in the account of the burning bush, how God said to him ... (Mark 12:26 NIV)
A more pertinent correlation with the declaration in John 8, is with a similar declaration made elsewhere in John, to the "Samaritan woman", likewise concerning the promise of messiah, in this case appealing to Jacob :
25 The woman said, “I know that Messiah” (called Christ) “is coming. When he comes, he will explain everything to us.”
26 Then Jesus declared, “I, the one speaking to you—I am he.” (John 4:25-26 NIV)
Jesus' declaration here - egó eimi - or "I am he" - a declaration to be messiah, is the same expression recorded of Jesus in John 8:58 variously rendered "I am".
This promise of a messiah or a deliverer, was given not only to Abraham in a figure in Genesis 22 and elsewhere, which promise is the subject of the conversation with the Samaritan woman, but also given before Abraham, to Adam and Eve in the garden.
And I will put enmity between you and the woman, and between your offspring and hers; he will crush your head, and you will strike his heel.” (Genesis 3:15 NIV)
This victory over serpent thinking, by a descendant of the woman, being echoed in a type elsewhere ...
Just as Moses lifted up the snake in the wilderness, so the Son of Man must be lifted up ... (John 3:14 NIV)
This promise covenanted with blood :
The Lord God made garments of skin for Adam and his wife and clothed them. (Genesis 3:21 NIV)
The "garments of skin" required an animal sacrifice, hence Jesus referred to as ...
... the Lamb who was slain from the creation of the world. (Revelation 13:8 NIV)
This promise or idea or logos being the subject of the opening passage of John.
First the promise of a redeemer - the idea or "logos", given in the garden, given afterward again, to Abraham and Isaac and Jacob, then the manifestation of the spirit of Elijah in John, who declares himself to not be the christ, then the manifestation or fulfillment of the promise in Jesus.
... God in his grace gave it to Abraham through a promise. (Galatians 3:18 NIV)
... Isaac and Jacob, who were heirs with him of the same promise. (Hebrews 11:9 NIV)
This logos and progression summarized thusly:
He who comes after me has surpassed me because he was before me. (John 1:15 NIV)
[ Proclaimed publicly at John 1:27, Matthew 3:11, Mark 1:8 ]
A proclamation by John, echoing Jesus declarations to the Samaritan woman, that he is the expected messiah, called christ, and to the Jews that he is the subject of the promises to Abraham and others prior. On both occasions responding in the affirmative - egó eimi ... I am [him].
19 But with precious blood, as of a lamb, unblemished and unspotted, of an Anointed One,––
20 Foreknown, indeed, before the foundation of the world, but made manifest at a last stage of the times ... (1 Peter 1:19-20 Rotherham)
29 On the morrow, he beholdeth Jesus, coming unto him, and saith––See! the Lamb of God, who taketh away the sin of the world.
30 This, is he, of whom, I, said––After me, cometh a man, who, before me, hath advanced, because, my Chief, was he. (John 1:29-30 Rotherham)
The clavicle being, "because my Chief, was he".