Hypostasis literally translated is substance, and that’s what Aquinas is referring to in this passage. For example, when he distinguishes hypostasis from subsistence he says:
For, as it exists in itself and not in another,
it is called “subsistence”; as we say that those things subsist
which exist in themselves, and not in another. As it underlies
some common nature, it is called “a thing of nature”; as,
for instance, this particular man is a human natural thing. As it
underlies the accidents, it is called “hypostasis,” or “substance.”
Literally speaking, the hypostasis is simply the underlying substance that is covered by accidents. In a red car, for example, the car would be the substance, and the red would be the accident (think of nouns and adjectives).
In Byzantine theological usage, hypostasis doesn’t mean “substance”, but rather means “this existing substance of a rational nature”. It came to have this meaning because “ousia” was already used to mean the general, underlying nature even though it technically just means the definition of a thing and not necessarily a really existing substance (so it would normally refer to “car” rather than “this car”). In theological developments of language, hypostasis (which, in the car example, would be the car which underlies the color red) came to represent what we call in English “person”, because a person is the highest and most perfect example of hypostasis.
Subsistence refers specifically to that which exists of itself, and can refer to anything from a rock to a Divine Person. In terms of Divine Persons, subsistence and hypostasis have the same meaning, in Latin and Greek respectively. In both cases the terms mean “this individual thing with this particular nature”. The reason that the equivalent of hypostasis (substance) isn’t used in Latin is simply because it came to refer to the general nature in Latin theological tradition (and, linguistically speaking that is a more precise usage), while the term subsistence carries the actual intended meaning of “hypostasis”, even though it’s not the literal translation.
Hope that helps!