I'm not going to get into a debate here, but Christianity doesn't believe in a Christian messiah. There is a Jewish messiah in the Tanakh and that's who they believe Jesus is. That doesn't mean he's the messiah of religious Judaism, but he (in that belief) is the Messiah prophesied in Judaism to the ethnically Hebrew figures in the Tankakh. I don't think anyone actually gets confused and thinks Jews accept him. It's obvious.
However, there never was a "Messiah to the Christians." Who were these Christians that Jesus supposedly preached to and was accepted by? There were no Christians at the time. Jesus lived in Judea, was Jewish, and was accepted by a certain number of first-century Jews as their Messiah. Others did not believe he was the Messiah. This was a division between one group of Jews and another. Present-day Christians are followers of one camp, present-day Jews are followers of the other, but they both share some common history, beliefs, scriptures, and terminology. It's not reasonable for one of those factions to tell the other they're not allowed to use any language that refers to the original faith.