There are generally 3 different interpretations of the prophecy found in Isaiah 53. The most dominant one which is the Christian interpretation of this passage of scripture outlined in the New Testament by the Apostles, is indeed attributed to Jesus Christ or Yeshua Ha-Moshiach, the Jewish Messiah. Perhaps the most famous part of the New Testament attributing this prophecy to Jesus is found in the book of Acts 8:26-40. The story about the Ethiopian eunuch who is sitting in his chariot and reading from Isaiah 53 when Philip approaches him as commanded by the angel of God. Philip asks the eunuch if he understands what he's reading. The eunuch replies, "How can I? Unless someone explains it to me. Who is the prophet talking about? Himself or someone else?" Then Philip starts with that very passage of scripture and tells him about the good news of Jesus. The eunuch then believes the gospel and asks Philip to baptise him.
The second interpretation is offered by the Jewish people who do not believe that Jesus of Nazareth was the promised Messiah. They accuse Christians of taking Isaiah 53 out of context and understand this servant to be the nation of Israel. The strongest argument made for this interpretation is that the servant is explicitly identified as Jacob or Israel in Isaiah 41:8, Isaiah 44:1, Isaiah 44:2, Isaiah 44:21, Isaiah 45:4, Isaiah 48:20, and Isaiah 49:3. This interpretation however falls apart pretty quickly. First of all, the first servant of the LORD is actually Isaiah himself in Isaiah 20:3. The second servant of the LORD is Eliakim, son of Hilkiah in Isaiah 22:20. Then in Isaiah 37:35 David is called the servant of the LORD. So clearly there's more than one servant in Isaiah's scroll and it's not wise to assume that just because Jacob or the nation of Israel is called the servant of God more than anyone else, therefore anywhere we see the word "servant" without an explicit identity, we must assume it's Jacob or nation of Israel.
Another reason why the second interpretation falls apart is that Jacob or the nation of Israel is considered a sinful nation in need of redemption whereas the servant of the LORD in Isaiah 53 is righteous and is likened to a lamb that is led to the slaughter for sin offering. Isaiah 42:18-25 calls Israel a deaf and blind servant who pays no attention, who's sinned against the LORD and has been handed over by the LORD to become loot. In contrast, there is another servant in the same chapter in whom the LORD delights, and puts His Spirit on; who will bring justice to the nations. In Isaiah 42:6, God says that He's going to give this servant away as a covenant for the people and a light to the gentiles. Isaiah 42:3-4 says this servant will remain faithful and will not falter, until he brings forth justice on earth. Sounds very Messianic if you think about it! Similar to the suffering servant in Isaiah 53, who is righteous and has no deceit in his mouth and dies not for his own sins, but for the sins of others.
The third interpretation offered by Rabbinic Judaism to solve this problem is that the servant is not the whole nation of Israel, but a righteous remnant within the nation of Israel such as the prophets. This interpretation fails as well for a number of reasons. Firstly, the servant is referred to in a singular fashion and if it's figurative language, then it must be the whole nation of Israel and cannot be a selection of it; but if it can be a remnant within Israel, then why can't it be the Messiah? Isn't the Messiah going be a the best representative of Israel? Also, all those references to Jacob or Israel to justify this interpretation is no longer binding because they are referring to the whole nation of Israel, not a righteous remnant within Israel. Secondly, Isaiah himself needs atonement for his sins in Isaiah 6:6, Isaiah 42:19 calls even the messengers sent to Israel deaf and Isaiah 43:27 says those God has sent to teach Israel have rebelled against Him!
In contrast to all of this, the suffering servant in Isaiah 53 is going to be highly exalted, which is a word used for the LORD Himself in Isaiah 6:1. The Hebrew word used in both is וְנִשָּׂ֛א [wə-niś-śā] which is best translated to extolled, highly exalted or lifted up. Furthermore, Isaiah 53:6 and 53:12 talk about how this servant makes intercession for sinners through his brutal death. Interestingly enough, Isaiah 59:16 tells us that God looked and couldn't find anyone to make intercession and so His own arm achieved salvation for Him. The Hebrew word used for intercession in these verses comes from פְגִּֽיעַ [p̄-gî-a‘] which is best translated as intercession. And fascinateingly enough, Isaiah 53 starts with: "To whom has the arm of the LORD been revealed?!" Isaiah 52:10 says: "The LORD will lay bare His holy arm in the sight of all the nations, and all the ends of the earth will see the salvation of our God." Let's not forget that salvation in Hebrew is Yeshua or basically the name of Jesus in Hebrew!
