Although there were a variety of beliefs among the ancient Jews as to who was being spoken of in Isaiah 53, a Targum (paraphrase) and the Talmud evidences that many understood this to refer to the Messiah:
“Behold, My Servant the Messiah shall prosper" from the Targum ("Targum Jonathan") to Isaiah 52:13, Exegesis of the Targum."
“The Rabbis said: His name is "the leper scholar," as it is written, Surely he hath borne our griefs, and carried our sorrows: yet we did esteem him a leper, smitten of God, and afflicted. [Isaiah 53:4].” -- Babylonian Talmud, Sanhedrin 98b