Jesus didn't claim he is the messiah. When Peter said to Jesus you are the messiah Jesus strictly commanded them “You should tell no one that he was Jesus the Christ". Jesus told the Jews not to believe any individual who claims he is the messiah. Not only that Jesus told the Jews no individual would come as the messiah as they expect, but who or what will come is the kingdom of God. 

Jesus told the Jews to always remain as Jews facing all the persecutions awaiting them and prepared to accept the Kingdom since it is not known when it will come. The Jews who expected Jesus is their messiah were disappointed that He didn't overthrow the Romans and declare himself King, like the Messiah was supposed to do. So they rejected Jesus saying he is a failed messiah and failed to understand his message. 

On the other hand, the Jews who found their messiah in Jesus interpreted words of Bible in such a way to make others also to believe Jesus is the messiah and son of God. The Jews who say Jesus is the failed messiah and the Jews ( later became Christians) who believe and preach Jesus is the Christ and son of God both went wrong in understanding the message of Jesus. 

The word messiah has no such meaning as “Savior “. Messiah simply means "Anointed".The Jews interpreted that word as savior and there they started the wrong interpretation of Bible. Yahweh is the savior of Israel and there is no other. Promise given to David and Salomon is conditional one. And sons of David many times failed to fulfill the condition. So God is not bound to keep his promise.But Jews inadvertently made the promise God gave to David an unconditional one and it led them to believe God is bound to provide them a super king “ Messiah” from the sons of David. This wrong interpretation of Bible words and belief in a Christ culminated in Christian religion.