The most common Jewish interpretation of that chapter was that it refers to the messiah; only in later medieval times in Western Christendom was an effort made to redirect the interpretation away from the messianic view (for defensive reasons).
But that the section refers to the messiah is not just a Christian interpretation, but also the traditional Jewish one.
(See, “The Fifty-third chapter of Isaiah according to the Jewish interpreters”, by Driver and Neubauer; an expensive pair of books, one in English, and one in the original languages — Hebrew, Aramaic, Arabic, etc.). possibly on line somewhere