Lie groups vs Lie monoids
There is a well developed theory of algebraic monoids, due principally to Putcha and Renner. I think Lie semigroups is less well developed but there is work by Hoffmann, Lawson and the thesis of Langlands was on this subject.
See the following:
MR1317811 Hilgert, Joachim; Neeb, Karl-Hermann Lie semigroups and their applications. Lecture Notes in Mathematics, 1552. Springer-Verlag, Berlin, 1993. xii+315 pp. ISBN: 3-540-56954-5 (Reviewer: Gestur Ólafsson)
MR1179336 Neeb, Karl-Hermann On the foundations of Lie semigroups. J. Reine Angew. Math. 431 (1992), 165–189. (Reviewer: Jimmie D. Lawson)
MR1235759 Mittenhuber, Dirk; Neeb, Karl-Hermann Remarks on our paper: "On the exponential function of an invariant Lie semigroup'' [Sem. Sophus Lie 2 (1992), no. 1, 21–30; MR1188629 (93j:22007)]. Sem. Sophus Lie 3 (1993), no. 1, 119–120.
Moreover, completions of infinite dimensional Lie groups (like diffeomorphism groups) with respect to right invariant Riemannian metrics tend to be semigroups, for an easy example see 4.8 of here.
Anders Kock mentions Lie monoids and some of their properties in his book on synthetic differential geometry. Basically, in SDG, a Lie monoid is a microlinear monoid object. It is easy to show that the tangent space at the identity of a Lie monoid is an $R$-Lie algebra, defined in the same way as for a Lie group, and isomorphic to the left- or right-invariant vector fields on the monoid (depending on your bracket convention). We also still have a Lie functor, taking Lie monoids to their Lie algebras, and Lie monoid homomorphisms to Lie algebra homomorphisms.
For example, for any microlinear space $M$ the mapping space $M^M$ is a Lie monoid, with the composition of maps as the multiplication. Then the Lie algebra of $M^M$ is the space $\frak{X}$$(M)$ of vector fields on $M$. Since each infinitesimal transformation is invertible, it follows that this is also the Lie algebra of $\text{Diff}(M)$.
One thing to note is that since we do not have an inversion map on a Lie monoid, then I do not see how would have a canonical isomorphism of Lie algebras between left- and right-invariant vector fields on the monoid. Usually the isomorphism would be given by pushing forward a left- or right-invariant vector field by the inversion map.