"Tamashii" doesn't mean judeo-christian Soul. It can be used in that way but it isn't exclusively that way and i would say for a Japanese person it wouldn't be the first thing they'd jump to. It is very often used to mean the emotion or spirit that drives a human. In fact I have a Japanese textbook which states that thinking of Tamashii as meaning "the spirit of the dead" is a mistake more likely to be made by foreigners than Japanese speakers. There are translations that have used it that way but they are contentious and unlikely to be used by a native Japanese spaker.
For example the english phrase Team Spirit, the Spirit there would be Tamashii. (And indeed a lot of sports animes use this term for example!) The claim that it is just for the immortal soul or even primarily for the human soul is erroneous. Trying to use Tamashii exclusively as Judeo-Christian Eternal Soul just doesn't work. If anything tamashii has a very heavily emotional aspect to it. It is very much considered to be 'what makes up a human.'
It literally means soul in Japanese just as kokoro literally means heart but that doesn't mean they translate over directly as such. There are also cultural differences at play here which make it even more complex because even though there is an idea of spirit and an idea of (emotional) heart in English they don't translate over directly. There are some differences in the core LR translation but I don't think the actual important context (B-man can take souls but he can't see inside human hearts) is missed.