Is there a generalization of the notion of braided monoidal category that does not force the braiding $\gamma$ to be an isomorphism? I mean, it is of course possible to define such a kind of category, but is this a common notion with an established name?
2 Answers
Day, Panchadcharam, and Street have a paper on lax braidings, though I don't think it could be called a common notion. Anyway, "lax" seems to be the obvious terminology to try here.

7$\begingroup$ *Brian Day, Elango Panchadcharam, and Ross Street. $\endgroup$– varkorCommented Dec 17, 2022 at 21:54
This generalization has also been studied by Victoria Lebed, which she called "prebraided" in her thesis Objets tressés: une étude unificatrice de structures algébriques et une catégorification des tresses virtuelles (title in French but most of the text is in English). Actually, Lebed motivates an analogous generalization of the notion of braided object in a monoidal category, defined as an object $A$ equipped with a (not necessarily invertible) morphism $\gamma : A \otimes A \to A \otimes A$ satisfying the YangBaxter equation. One advantage of dropping the invertibility assumption is that it permits studying idempotent braidings, in particular see her papers:
 Cohomology of idempotent braidings, with applications to factorizable monoids
 Plactic monoids: a braided approach
for some very interesting examples of idempotent braidings in $(\mathbf{Set},\times,1)$.