Question Restatement: Is it correct to call the interpretive practice of injecting "Jesus" into old Testament prophecies, (after the fact), "Eisegesis"?
Question 2, (#1 Reworded): Were New Testament author's practicing "Eisegesis", by injecting Jesus into prophecy?
Catholicism & Judaism's Eisegetical Interpretation of Prophecy
Anyone who has ever read Radak's commentary on Psalms 110, "the Psalm ABOUT David ...", would immediately understand that Judaism relies on Tradition, and therefore Eisegesis, to Interpret Prophecy.
Answer: Controversially, in Catholicism, and Judaism, Tradition, and therefore a Rabbi, are both necessary to interpret prophecy--which is certainly Eisegesis. See Maimonides' Introduction to the Talmud, (Chapter 1 & 2, Prophecy). (I cannot find the reference regarding Catholicism.)
Christianity, and the Exegetical Interpretation of Prophecy
This is an issue between Theory and Practice, (and it is impossible to characterize modern practice--objectively). Some do one or the other, or both; others certainly do not.
Answer: "In Theory", Regardless of the accuracy of Christian interpretation/translation of prophecy, and regardless of Tradition, Christian texts admonish against Eisegetical interpretation of Prophecy, and encourage Exegetical Analysis.
The problem is, now being so far removed, it is almost impossible to do this Exegetically--which is probably why the Christian warning against Eisegesis exists :
2 Peter 1:20, NASB - But know this first of all, that no prophecy of Scripture is a matter of one’s own interpretation, 21 for no prophecy was ever made by an act of human will, but men moved by the Holy Spirit spoke from God.
Since there IS textual basis to assert that the prophets, themselves, encountered the "Messiah", then "Deductive Inferences" can be drawn from the texts--which is Exegesis. And, since the Apostles are claimed to be "Witnesses", taught by the Messiah, from Prophecy, himself, this would "theoretically" remove their writings from the realm of Eisegesis as well, as they would simply just be recording what they heard.
NOTE: The interpretation of Pre-Messianic encounters can also be exegetically drawn from the texts, (i.e. Moses, David, Isaiah, Jeremiah, Zechariah, Daniel, etc). Notably: Judaism sometimes reconciles this with the identity of "Metatron, (Link)".
When Jesus and Paul spoke of interpreting the prophets, they both pointed people to the texts, and directed them away from Tradition and hear-say.
Luke 24:25, NASB - And He said to them, “O foolish men and slow of heart to believe in all that the prophets have spoken!
Acts 17:11, NASB -
Now these were more noble-minded than those in Thessalonica, for they received the word with great eagerness, examining the Scriptures daily to see whether these things were so.
Jesus' Exegetical Approach to Psalms 82:6, Nearly Got Him Stoned
John 10:34 - Jesus answered them, “Has it not been written in your Law, ‘I said, you are gods’? 35 If he called them gods, to whom the word of God came (and the Scripture cannot be broken), 36 do you say of Him, whom the Father sanctified and sent into the world, ‘You are blaspheming,’ because I said, ‘I am the Son of God’?
NOTE: Ironically, Christians explain away Jesus' comment, by Eisegetically injecting "Sarcasm" into his statement.
NOTE: There many other "Exegetical"/"Eisegetical" concepts in the texts--if there is interest.