In the Summa Theologiae or Summa Theologica, Thomas Aquinas outlines distinct senses for the noun phrase "subsistent thing" (i.e., this particular thing). I guess that ST I.75.2, Reply to Obj. 1 touches on the question I used to have when reading Aquinas. He there maintains that "this particular thing" can be understood in two senses: a) "for anything subsistent" and b)"for that which subsists, and is complete in a specific nature." The soul (according to Aquinas) is "this particular thing" in the first sense, but evidently not in the second. So he does not conceive of the soul as subsisting in a Cartesian manner. Descartes memorably refers to himself as a "thinking thing" (res cogitans) yet it doesn't seem that Aquinas would identify the person with the soul like the French Catholic Descartes does.
Kevin Corcoran parses the topic in Rethinking Human Nature: "Here is one way to mark the difference between Aquinas and Descartes. In both views, there are two 'things.' In Descartes' view, the two things are complete substances (soul and body). In Aquinas' view, there are two incomplete substances (soul and matter) that jointly compose a complete substance (a human being)." See Corcoran, page 38.
This observation is based on ST I.75.4: "Not every particular substance is a hypostasis or a person, but that which has the complete nature of its species. Hence a hand, or a foot, is not called a hypostasis, or a person; nor, likewise, is the soul alone so called, since it is a part of the human species."