Well, god is quite falsifiable actually, otherwise its being true would have any effect on the world and it would be ruled out by Ockham razor. Believers have to state things like "he created the Earth in 6 days". Once proven false, is falsified. And it was.
The odds with the dodo are different in terms of how likely it is that an animal is not extinct (happened in the past, you can figure out an easy way for it to happen, would fit in what we know is true and so on) vs the whole universe is created by a single entity and so on, which has a huge attack area, has never been proven true, and would require a reshaping of everything we know. This makes it less likely.