Yes, Jesus knew who he was as the following facts show.

Jesus did not often speak of his eternal past; he apparently deliberately did not say much about it. He however said enough to let us know who he was and that he knew who he really was. His memory reached as far back as Abraham and time before the world began.

As you consider these facts, please take note that some of them cannot be explained in any other way other than he actually remembered that past. Please note that Jesus was not making these claims by faith.

1. Jesus knew at the age of twelve that he was the Son of God for he referred to God as his Father [Luk 2:49] The miraculous power of Jesus in his later years show that he wasn’t just an intelligent or deluded young boy, but that he was indeed what he claimed to be as a child—the Son of God.
2. Though uneducated [Joh 7:15], Jesus knew the scriptures thoroughly at the age of 12 [Luk 2:42-48]. Where could he have gotten such wisdom from at that age and without education?
3. Jesus deliberately took Peter, James and John up to the mount of transfiguration to show them his glory [Mat 17:1-13, Mat 16:27; 2Pe 1:16-18]. We can learn from this that he could have at any time returned to his previous state of glory (apparently then, the big challenge was accomplishing the same fit in death)
4. Jesus’ conversation with Moses and Elijah on the mount of transfiguration [Mat 17:1-3; Luk 9:28-31] also shows that Jesus had knowledge of who he was before coming to earth. The accounts do not present their meeting as something extraordinary for Jesus but as normal.
5. Jesus spoke of his personal meeting with Abraham thousands of years before and that he existed before Abraham [Joh 8:56-58]. Obviously, he remembered his eternal past.
6. Jesus remembered what the Father looked like [Joh 5:37; 8:55; 6:62]
   “No one has seen the Father except the one who is from God; only he has seen the Father.” [Joh 6:46]
7. Jesus knew where he came from, that he came from heaven
   “Jesus answered, ‘Even if I testify on my own behalf, my testimony is valid, for I know where I came from and where I am going. But you have no idea where I come from or where I am going.’” [Joh 8:14]
8. He prayed that the Father would restore to him the glory he had with Him before the world began [Joh 17:5]. Clearly, he had memory of who he was before the creation of the world.

A Contradiction To Knowledge Of His Previous Life

One of the main contradictions to the conclusion that he knew who he was etc., is the fact that the Scripture say that he grew in wisdom [Luk 1:52][7]. One may wonder, how can Jesus grow in wisdom and still we claim he retained knowledge from his former life.

The Human Nature And The Limitation Of Knowledge

Though Jesus was not wiped clean of the memory of his previous existence, we must note that, practically speaking, he couldn’t have remembered everything. This is because part of the limitations of mortal, human nature is that its knowledge is not exhaustive.

In Isaiah 40, where God contrasts his glory or nature against human nature, we learn that part of the characteristics of God’s glory or divine nature is his knowledge, his ability to remember everything—all the dust in the world, all the water, the weight of the mountains, etc [Isa 40:12].

Lift your eyes and look to the heavens: Who created all these? He who brings out the starry host one by one, and calls them each by name. Because of his great power and mighty strength, not one of them is missing. [Isa 40:26ff].

Do you not know? Have you not heard? The LORD is the everlasting God, the Creator of the ends of the earth. He will not grow tired or weary, and his understanding no one can fathom.  [Isa 40:28]

Angels too are described in scripture as having near perfect knowledge [2Sa 14:20][8]. Though, their knowledge is evidently not equal to that of God, it is far greater than that of humans.

Therefore, since the human nature is limited in the knowledge it can possibly possess, Jesus in human nature could not have had a perfect recollection of everything he ever did or was. This, however, is only true without the power of the Holy Spirit.

With the power of the Holy Spirit who brings all things to our remembrance [Joh 14:26][9], Jesus could recollect everything that he was and knew before his humanity. Thus, before Jordan, Jesus would certainly have had recollections of his time before his humanity but because of his human nature, without the power of the Holy Spirit, he couldn’t have remembered everything and every single detail.

The Scriptural Difference Between Humanity And Divinity

Glory is what differentiates the divine from the non-divine. In very simple terms, what makes one different from the other is the nature of the body.

In practice, the basic differences between the two natures are as follows (please see Isa 40 where God compares the human and divine natures):

1. Strength, endurance—(the divine is unlimited [Isa 40:28] human strength and endurance, even in its prime, is limited [Isa 40:30-31])
2. Wisdom [Isa 40:14], knowledge [Isa 40:12-14], understanding [Isa 40:28],
3. Longevity—(the human nature fades and perishes [Isa 40:6-8,], the diving nature never perishes [Isa 40:8,28]),
4. The divine nature has extraordinary abilities such as the ability to fly [Isa 40:30-31]

It is impossible to be both natures at the same time. It is impossible to be weak and strong, mortal and immortal, earthly and heavenly in every sense at the same time. One may only be weak and strong at the same time if they are in different senses but not if they are in the very same sense. Thus, we can see that it is impossible for Jesus to be 100% human and 100% divine in every sense at the same time.

This is the truth of the Gospel and Holy Scriptures and not a form of Gnosticism or some other heresy; Jesus was not 100% of both natures (human and divine) at any time—he had to be human to atone for our sins and had to be divine to enter heaven.

Summary

Jesus was bodily human, he dwelled in our human body of flesh and blood and this affected his knowledge as it did other things such as his physical strength [Joh 4:6]. When the Holy Spirit came on him however, he had full recollection by the power of the Holy Spirit who brings to our remembrance all things.

Conclusion

All that was needed for the atonement and for Christ to be our high priest under the New Covenant was for Christ to share in our nature. It is not required for him to be void of knowledge of who he was; whatever quality or attribute we may regard as divine which he lost as a human being was lost not as a requirement but as a natural consequence of taking on human nature.

Jesus was bodily human, he dwelled in our human body of flesh and blood, Jesus was a 100% human. He shared in every common natural human weakness. This meant that he was physically and mentally not as strong as angels or as God which he was before becoming human. Although he retained all his divine rights as the Son of God, as for example, he could have returned to his previous glory at any point in his time on earth as fully demonstrated on the mount of transfiguration, he was physically human. The Holy Spirit however made a big difference in him.

The Holy Spirit imparts extraordinary abilities to ordinary men. He gave Samson extraordinary strength; Moses extraordinary health and age; Joseph, Solomon, Daniel, etc., extraordinary wisdom and so on. The extraordinary abilities the Holy Spirit imparts to a person however does not change the fact that they are mortal and human.

Jesus received with the Holy Spirit extraordinary abilities, such as a sound memory, but as with other men gifted by the Holy Spirit, he was still mortal. Through the Holy Spirit who brings to our remembrance all things, he had full knowledge of who he was from eternity past.