Why do Christians believe that the gods of the Old and New Testaments are the same?

They're confused.

It's obvious that they are different canaanite gods.

In the Old Testament, God showed His personality. But there were also times when Jesus showed up as "AngelOfTheLord". He's got the badass attitude (like stopping Balaam with a sword) but also kind enough (like when Gideon asked for a sign).

In the New Testament, God made Jesus the substitute for punishing people, when before, the people should DREAD God's punishment towards them. Jesus became the righteousness of the soon-to-be followers (also called Christians). Hence, the emphasis for love, joy, peace etc.

There is only one true God in the Bible. In the Old Testament people worshiped a lot of false Gods.

Indeed, over the thousand years or so that these writings were collected, ideas about God changed. They had to, because once you decide to do without the usual squabbling pantheon and settle on one deity, you can't play gods off against each other any more. And so that one God had better be worthy of worship in every possible way.

The development took place gradually. If you think there is just one depiction of the "god of the Old Testament" then you haven't bothered to read much of it. The notion of God in the Torah is changed many times, by Isaiah, Jeremiah, the Job poet, Amos, David's lyrics, and the satirist who wrote Jonah, among others.

The development continued, beyond the time when the last parts of the Old Testament (which vary, depending on which Old Testament you're using) were written. Jesus picked up from much more recent developments than the ones we see in the OT. The "Great Commandments" were a popular distillation of the central principles of older teaching: notice that Jesus obtains a recitation of them in Luke 10, by a lawyer who argues with him. A different formulation of the Golden Rule had already been propounded by Hillel.

People don't think the same as their predecessors, a thousand years of history before them. They don't even use their language the same way. (We're talking over twice the time between us and Shakespeare, you realize.) And the New Testament was even written in an extremely different language, in which it was very difficult to express the same ideas exactly.

In that respect, their beliefs about God had to be different. But they still identified their God by one specific relationship: they worshiped the God of Abraham.

Gods?

You mean the Triune God...?

God's actions in history and creation are executed as the respective works of the Father, the Son, and the Holy Spirit. Nothing has changed.

How can it be the same god?

God is supposed to be omniscient, why would such a god need a new covenant? Unless He was just messing with us.

Most people and Christians know that there is one God dear. Your question is simply inspired by fundamentalistic and dogmatic atheism, which needs to create divisions as it can't stand on it's own. It's as ridiculous as it sounds dear.

It's the same God...there is and always was, only ONE.

The old and new testament is all apart of one full book - the Bible. Therefore, the gods and true God recorded in it is the same referred to in both portions of the full book.

God did not change

Our relationship changed. at the cross

We entered a "Grace Period" where God's Wrath was forestalled, giving SOME of us the opportunity to join with God as Children

No worries, if you miss the old attitude, it will be back in full force, on Judgment Day