My knowledge and expertise are certainly not profound enough to answer this question as a real scholar would do. However, let me answer with an in-your-face provocative counter-question:
Is there a distinct "Amalekite" soul? So to say: is a soul bound by its alleged nature to a destiny? Is the free will shadowed by a predestined nature of our soul which we will follow anyway?
The mystical discussion about the soul's essence is interesting. It can, however, become a rather dangerous terrain. To state that there is a distinct Jewish soul bound and attracted by its nature to the Torah's righteous wisdom implies that there must also be the counterpart of such entity, i.e. a soul who is destinated to contrast the divine values and those who represent them. An extreme conclusion of such a scenery would be that there is no right nor wrong. No matter how righteous or wicked you are, no ultimate justice is needed since you ultimately only follow the ways of your soul in a cruel play dictated by a higher entity. I would have no desire to be part of such world.
What if a so-called Jewish soul feels the need to explore other spiritual paths or if such a soul turns against its own "soul group" and begins to contrast the Jewish values? What if a soul that follows the ways of Amaleq comes to realise how futile this uphill battle is and begins to explore the paths of righteousness?
The mystical thought elaborates largely the idea of a Jewish soul vs. a gentile soul reaching ideas that can be described with the r-word; many of these ideas have brought really bad reputation to the Jewish theology. Take for example the end part of the 1st chapter of Likutei Amarim in Tanya:
כי בישראל נפש זו דקליפה היא מקליפת נוגה, שיש בה גם כן טוב... מה שאין כן נפשות אומות העולם הן משאר קליפות טמאות שאין בהן טוב כלל... כמו שכתוב בעץ חיים שער מ״ט פרק ג׳: וכל טיבו דעבדין האומות לגרמייהו עבדין
(For in Israel the soul k'lipah comes from the k'lipah called nogah, which also contains good...The souls of the nations come from the unclean kelipot which have no good in them whatever...as is written in Etz Chayim, Portal 49, ch. 3: and all the good that the nations do is out of selfish motives.)
This daily Tanya lesson can be found here on Chabad.org
As long as the path of the Torah is seen as The Truth with a specific soul type bound to it by nature the division between Jewish and non-Jewish souls would imply that there is no freedom of choice, no progress, no justice/injustice, only a predestinated play where you are born, live and die as a helpless puppet driven by an uncontrollable instinct. Such a scenery would mean an endless downfall and finally to be condemned to Gehinnom for an unsuccessful "Jewish soul" who cannot fulfil the Torah's requirements he's bound and that he/she cannot get rid of. Such a scenery would mean for the other souls that the "Torah is indeed beoynd the ocean and I cannot do anything to reach it".
I have never heard of any definition of a "Jewish soul" in the Hazal. Please inform me if I've missed something.