In the Qur'an, when used alone, the word "nafs" is not really used as a synonym for soul (in the sense of the soul as a non-corporeal entity distinct from the body). Rather, it is usually used to refer to the complete self; for instance, in phrases like "bring yourselves" or "they deceive themselves".
(Of course, this distinction can be blurry based on how one perceives the nature of the soul and the relationship between the soul and the body.)
This is different from the use of the word "nafs" outside the Qur'an, in that "nafs" in other texts or discussions is often used to mean specifically the immaterial part of the human.
Therefore, when the Qur'an says "kullu nafsin dha'iqat al-mawt", it means that all people will taste death in a holistic sense, not specifically that all souls will.
However, since the soul and body experience things simultaneously, the soul will also taste the experience of death, just as it experiences other things occurring via the body.
If one understands death to mean "transfer from realm to realm", rather than "annihilation", then definitely the soul will experience transfer from realm to realm upon death.
Some say that we will have a different type of body as a vehicle for the soul in the barzakh and in the Hereafter. Therefore, if death is also understood to mean "transfer from one type of body to another", the soul will also taste death in that way.
So it is not incorrect to say that all souls will taste death; the usage of the word in the Qur'an is simply more expansive than that.
Note that compound phrases in the Qur'an such as nafs al-'ammarah (the bestial soul), nafs al-lawwamah (the blaming soul), and nafs al-mutma'innah (the content soul which God calls to Paradise) refer more to immaterial aspects of the human being.