But for the mistakes pointed out by Noah below, all of this (and much more) is found, e.g., in the 3rd chapter of Golan's Semirings and their Applications.

Addenda. Even replacing, as would be reasonable, 'ring' with 'semiring' and 'bimodule' with 'bi-semimodule' in the phrasing of the previous comment.

@Yemon. OK, there is an analogy, but it's quite feeble, I think. I mean, I don't see any way to link the Dorroh extension, say ${\rm Drh}(\mathbb A,\mathbb M)$, of a unital ring $\mathbb A$ by a $(\mathbb A,\mathbb A)$-bimodule $\mathbb M$ to the unitization $\mathbb S^{(1)}$ of a certain semigroup $\mathbb S$, as defined by (i), in terms of the existence of a semigroup isomorphism between $\mathbb S^{(1)}$ and the multiplicative monoid of ${\rm Drh}(\mathbb A,\mathbb M)$ for an appropriate choice of $\mathbb A$ and $\mathbb M$. If this isn't the case, then I wouldn't rely on such terminology.

@Qiaochu: In that case, the very same article by P. Freyd mentioned by A. Mathew in his post at amathew.wordpress.com/2012/01/26/homotopy-is-not-concrete, includes another example (apart from the usual category of topological spaces and homotopy classes of continuous functions).

The question makes no sense to me. In NBG, and axiomatic systems with a similar ontology, a category is a tuple $(\mathcal C_{\rm o}, \mathcal C_{\rm h}, s, t, c, i)$, where $\mathcal C_{\rm o}$ and $\mathcal C_{\rm h}$ are classes (and the members of a class are sets).

@Frank Stephan. Sorry for the delay in replying. This doesn't count as an example. The OP refers to ordered sgrps, termed linearly ordered sgrps, where strict inequalities are compatible with the composition law.

@Igor. This would confirm what I've been ever told about Lang's "idiosyncrasy to categories", wouldn't it?

Thank you, Robert. Somehow, following the links in the Wiki article I ended up with a discussion relating to the same issue from the Cat mailing list: permalink.gmane.org/gmane.science.mathematics.categories/734‌​. There, Peter White confirms (based on Colin McLarty) that the term was first coined by Norman Steenrod, while Michael Barr reports that Paul A. Smith said (concerning General Theory of Natural Equivalences) that "[...] he never read a more trivial paper in his life."

Of course, I did. On the bottom of page 4 in Monads on composition graphs (in "Applied Categorical Structures", Vol. 10, 2002), it is Schröder himself who claims that the definition of a composition graph given in the paper is somewhat informal (and I must really agree with him), while inviting the interested reader to look at his 1999 PhD thesis for something more formal.

A friend of mine sent me a copy of the paper right today, based on the suggestion of @quid. Thank you all the same for your helpfulness, @Seva.

Indeed, my question originates from Hegyvari and Hennecart's paper: hegyvari.web.elte.hu/Freimanmodels.pdf, which in turn is inspired by Green and Ruzsa's work. I've given a read of Tao's paper. It is definitely focused on measure-theoretic/metric/entropic analogues of (many) results from the theory of sum-sets but still in the (non-commutative) setting of groups (as far as I'm concerned, I don't like much the notational distinction between the commutative and non-commutative case). Freiman morphisms seem to enter the scene in Sect. 7, but they are not even mentioned explicitly.

Thank you, quid. I will try at the ENS (somehow, it didn't appear among the results of my search).