Firstly, we are dealing with religion, coherency and sense are not mandatory.
Secondly, the question you have put before us is far more complicated than you may think. Not all factions of the Jewish or Christian faiths agree internally with each other on the nature of god, let alone with a seperate external faith.
In theory, Jews, Muslims, and Christians are all praying to the same skygod, they just disagree on the details. At some point in their history a faction within the Jewish people decided to write their version of events, giving us the Torah, to which other writings were added, giving us the Tanakh. When Jesus made his big debut he was talking to Jews about Judaism; he was essentially a Jewish reformer. He did not set out to start a new faith, just revise the one that was already there. So Christianity began as a Jewish cult. But people were really fond of this carpenter, and soon his message eclipsed that of the Tanakh, and people started writing what he said down, long after he had died. There were many versions floating around, some of which were collected into a 'New Testament'; the rest were suppressed. Later, an Arabian trader heard about this amazing new pyramid scheme called 'monotheism' and decided to start his own franchise, picking and choosing the bits he liked, and adding his own personal spin.
For Christians the NT is a continuation of the OT; the OT promises a messiah, and the NT is about how that promise is fulfilled. The god is the same in both, only that in the sequel there is a new sidekick, so there is no reason why Christians wouldn't claim the god of the OT to be the same as the one in the NT; for them it is a logical sequence. Furthermore, the notion of the trinity is not explicitly used in the bible, it was developed a few hundred years later, though it is hinted at. And theologians have built imaginary empires on the merest of hints.
Your statement "The Jews of the Old Testament do not believe that Jesus is God, not even close" is puzzling; of course they wouldn't. If you are speaking historically, of the Jews in the Old Testament, well, Jesus wasn't born until hundreds of years after the Tanakh was written. If you are referring to modern Jews, why would they believe in Jesus, to them he was delusional faith-healer who started a really annoying cult that ended up persecuting them for the best part of two millennia, and they are content with their own firmly-established religion. That said, some Christians do not believe that Jesus is God, but is filled with divinity, or has a dual nature, etc. etc., which is one of the reasons why there are so many flavours of Christ-lovers in the world.