Parmandur
Book-Friend, he/him
I think we should take a step back from the Tiamat = Takhisis argument to get some more perspective.
There have been plenty of hints/innuendo that even when the gods appear on different worlds, they often don't manifest in the exact same ways. Meaning, a god can be worshipped on several different worlds, but because they usually aren't worshiped by the exact same type of people in the exact same way, their aspect/avatar on each world will manifest in different ways.
Meaning, practically Tiamat and Takhisis are the same god; the same way that the god in the Torah and the Bible is technically the "same god." But the worshipers don't really agree they're the same (same in real life), which is why for example they have different names, and interact with the world differently, have different goals.
For example, in Dragonlance, Takhisis' is married to the god Sargonnas, who is not mentioned in a non-Dragonlance world. Tiamat in other worlds is mentioned in a constant rivalry with Bahamat, who is not mentioned in Dragonlance.
So, is the underlying god the same, the one who dwells in the Nine Hells? Sure, I guess she can be the same. But the manifestation on each world is going to be different, so in practical terms they're not the same.
TLDR: Everyone is right, stop arguing.
They are also make-believe game objects.