Yes, but what’s particularly odd about a belief in the soul?

It depends what exactly you mean by 'soul'. If it's some sort of disembodied manifestation of our personality and memories that exists indepedent of the body then the problem is that there's no evidence for it. And instead we find evidence supporting (but not explaining) that our 'being' is one and the same as our brain. Consider, for example, that damage to specific parts of the brain not only impair certain parts of personality but can also destroy specific memories. That's because these things are physically one and the same as the brain. We just don't understand the subjective concious experience but there are some pretty solid boundaries delimiting what it is not.

If I lived before the development of our modern understanding of neuroscience and the invention of computers, I would not have enough evidence to be convinced that it was false even in our universe

I agree. In prehistory I would have believed the breeze in the crops was the work of elves and that thunder was sent by Thor because that's what it looks like and they didn't know better. But we don't live there. And we do know better.

but presumably souls would be assigned to bodies some time before birth

by who?

There's no evidence for this, of course. If one wants to pick the Judeo-Christian tradition there's still no reason to believe it given the bible that Jesus used considers early foetuses to not be people.