The ideal class semigroups mentioned in the question got studied in this setting (orders of number fields) and in other and more general ones by various authors in recent years.

A starting references is:

Zanardo, P.; Zannier, U. The class semigroup of orders in number fields.
Math. Proc. Cambridge Philos. Soc. 115 (1994), 379–391.

They show that this semigroup is a Clifford semigroup for orders of quadratic fields, *but* for degree greater 2, there always is some order such that the respective class semigroup is *not* a Clifford semigroup.

There is also an older paper on this theme (that it appears was overlooked in recent literature for some time but reapperas in still more recent one)

Dade, E. C.; Taussky, O.; Zassenhaus, H. On the theory of orders, in paricular on the semigroup of ideal classes and genera of an order in an algebraic number field. Math. Ann. 148 (1962) 31–64.

Starting from these two papers and looking for papers that quote them in MathSciNet for example one will find several more recent contributions. Investigating these semiclass groups; but it seems (but I donot have a good overview) the emphasis is more to generalize to more general structures (say, other domains than just orders) than more details in the number-theoretic setting.

Yet, in particular the early papers (I do not know for the recent ones) contain also some explicit information on these semigroups, but as commented earlier I would not know of in some sense simple descriptions (but again my knowledge is superficial here).