You're confusing a nefesh elokit(G-dly soul) with a nefesh Yehudit(Jewish soul). I have yet to see a description in Tanya of a nefesh Yehudit. Tanya is describing a nefesh elokit (G-dly Soul), a nefesh haAdam (human soul), and a nefesh bahamit (animalistic soul).
The nefesh haAdam(human soul) which everyone has, if it loses itself completely in the study of Torah and "grasps" Torah as defined by Tanya, then the seichol(intellect, which is the upper 3 attributes/sefirot) grasp the inner essence of G-d and therefore grasp a nefesh Elokit (G-dly soul).
All humans who have the nefesh haAdam(to my knowledge) can grasp a nefesh Elokit. Non-Jews should only grasp their seven mitzvot at first if they so desire, but maybe more to complete the seichel(one should be talking with an orthodox Rabbi at this point). Also, one cannot fully grasp the 613 organs of "the King" as Tanya describes, without fully grasping the 613 mitzvot.
The translations that it's a "Jewish soul" is wrong. It's a G-dly soul, not necessary a Jewish soul that the Tanya describes. All nefesh Yehudit(Jewish souls) are nefesh Elokit(G-dly souls), but not all nefesh Elokit(G-dly souls) are nefesh Yehudit(Jewish souls).