“You are the Messiah, the Son of the living God”Painting by Tissot
Q: Why did Jesus never refer to himself as the Christ, Messiah, son of God, but as son of man, teacher, rabbi?
A: He did refer to himself that way when asked point blank: “The high priest asked him, ‘Are you the Messiah, the Son of the Blessed One?’ ‘I am,’ said Jesus,” (Mark 14.61–62 NRSV).
Another instance: “[Jesus] asked his disciples, Who do people say the Son of Man is?… “Simon Peter answered, “You are the Messiah, the Son of the living God.” (Matthew 16.13 - 20 NRSV), which Jesus accepted, blessed Peter for his insight, and then told his students not to spread it around.
Some translations say "Christ" in these 2 passages but “Christ” is just the Greek translation of “Messiah.”
“Son of Man” originally meant “a person,” but by Jesus’ time it had taken on messianic connotations because of the episode in the 7th chapter of Daniel where “one like a son of man” presents himself to “the Ancient of Days” (i.e., God) and is “given authority, glory, sovereign power” and is “worshiped.” In fact, Jesus alludes to this scene right after confessing to being the Messiah in the Mark passage I quoted above.
“Son of Man” also had other advantages. For instance, it let Jesus refer to his messiahship indirectly so as to maintain a measure of modesty. Plus, “Messiah” had a lot of militaristic baggage to it. The common people expected a warrior like the ancient King David when they used the term. But “Son of Man” was a more mysterious figure from prophecy and less commonly used. This gave Jesus the opportunity to define the kind of Messiah he was instead of having current assumptions forced on him.
And of course, part of his idea of a Messiah involved being a teacher and a rabbi to his people.