The Legitimacy Of The Mark 15:34 And Matthew 27:46 Accounts Of Jesus' Words

Typically, this is seen not as an indication that Jesus felt abandoned by God. The sentence is a quote from Psalm 22:1.

My God, my God, why have you forsaken me?
Why are you so far from saving me,
so far from my cries of anguish?

This might seem to indicate that Jesus felt himself to be abandoned. But the psalm continues from feelings of abandonment to being supported by God:

But you, Lord, do not be far from me.
You are my strength; come quickly to help me.

Deliver me from the sword,
my precious life from the power of the dogs.

...

I will declare your name to my people;
in the assembly I will praise you.

You who fear the Lord, praise him!
All you descendants of Jacob, honor him!
Revere him, all you descendants of Israel!

For he has not despised or scorned. the suffering of the afflicted one;
he has not hidden his face from him. but has listened to his cry for help.

This is certainly something appropriate for someone with complete trust in God to say in a dark hour.

To very briefly address first your question about what language was Christ speaking versus what language Scripture was written, it is fairly obvious that Aramaic, Greek, and Hebrew were known at that time. The language question may be the OP reasoning to explain the more important question about Christ saying why have You abandoned Me God? Why would Jesus say that?

The saying is a quote from Psalm 22:1. Christ is on the cross. He is not coming down.

My God, my God, why hast thou forsaken me? why art thou so far from helping me, and from the words of my roaring? KJV, Masoretic Text

O God, my God, attend to me: why hast thou forsaken me? the account of my transgressions is far from my salvation. Septuagint

He also knows this fulfillment from verse 18.

They part my garments among them, and cast lots upon my vesture.

Christ dies on the cross and is buried.

But in the same Psalm at verse 22, there is a verbal transition from "what is" (perfect aspect) to "what will be" (imperfect aspect). Christ knows this too. Notice the "will".

I will declare thy name unto my brethren: in the midst of the congregation will I praise thee. Masoretic

I will declare thy name to my brethren: in the midst of the church will I sing praise to thee. Septuagint

So, to answer the OP, we don't need to delve into our covenant and how it was made or Christology questions to generally understand what was happening during the death, burial, resurrection. Christ who was perfectly righteous was sacrificed and because He was righteous was raised from the dead.

Heb. 10:12 But this man [Christ Jesus], after he had offered one sacrifice for sins for ever, sat down on the right hand of God;

Mark begins his book by drawing attention to the words of Malachi.

Behold I send my messenger ... Malachi 3:1

But there are two messengers in the quotation that follows. The messenger of preparation (who we know is John the Baptist) and the Messenger of the Covenant whom Malachi expressly states is :

The Lord whom ye seek.

Since the Lord himself, come in flesh, is the Messenger of the Covenant thus Mark quotes his exact words (in Aramaic) - Eloi ! Eloi ! - since that is what his book is about - Jesus Christ come to express the Message, the Everlasting Covenant of the New Testament.

But Matthew draws attention to the King (of the Jews) and the Kingdom (not of earth but of heaven). Thus Matthew's focus is the transition from the passing phase of limited revelation within the compass of the nation of Israel into the full revelation to the whole earth through the Apostolic word of the Everlasting Gospel.

Thus Matthew draws attention to the Hebrew scripture. He does not quote the dialect in which Jesus spoke - he quotes the Hebrew words from the Psalm of David the King - Eli ! Eli !

My God, My God, why hast thou forsaken me ? ... Psalm 22:1

And why was it so ?

How could God ever forsake his servant in whom he delighted ?

Because God spared not his only begotten Son, but delivered him up for us all. That Divine Righteousness might be satisfied, sins must - they must - be resolved in humanity. Not the unclean and compromised humanity of Adam or any of his sons, but in Another Humanity, come of woman, but not come of Adam.

Thus the shocking rift in Deity, between God in heaven and God in humanity. That perfect unity of Father and Son - in One Divine Spirit - was disrupted. And for whom ?

For a vast multitude - as the sand of the sea and as the stars of heaven - whom no man can number.

I believe this cry came from the crucified Jesus spontaneously in the depths of unimaginable suffering when he experienced that appalling separation from the Father which was essential to the resolution of the sins of men.

And I believe that David was granted to prophesy of that moment in his Psalm.

Jesus is not - actually - quoting the Psalm, I would say.

It is the Psalm which, a thousand years before the event, is quoting Jesus.

Jesus cried out (spontaneously) in his native dialect of Aramaic.

David (the King) records it, in spirit, in prophecy, in Hebrew.

Thus Mark notes the actual cry of the Messenger.

And Matthew quotes the Hebrew Psalm of David.