Of course the real question is whether abelian groups are really more fundamental objects than commutative monoids. In a sense, the answer is obviously no: the definition of commutative monoid is simpler and admits alternative descriptions such as the one I give here. The latter description can be adapted to other settings, such as to the 2-category of locally presentable categories, which shares many formal properties with the category of commutative monoids (such as being closed symmetric monoidal, having a zero object, having biproducts). As such I would claim that any locally presentable closed symmetric monoidal category is itself a categorified version of a semiring, not in the sense you describe, but in that it is an algebra object in a closed symmetric monoidal category, so we may talk of modules over it, etc.
However, it is undeniable that there is a large qualitative difference between the theories of abelian groups and commutative monoids. Observe that an abelian group is just a commutative monoid which is a module over $\mathbb{Z}$ (more precisely a commutative monoid has either a unique structure of $\mathbb{Z}$-module, if it has additive inverses, and no structure of $\mathbb{Z}$-module otherwise). The situation is analogous to the (smaller) difference between abelian groups and $\mathbb{Q}$-vector spaces. I do not know of a characterization of $\mathbb{Z}$ as a commutative monoid that can be transported to other settings. It seems that there is something deep about the fact that $\mathbb{Z}$-modules are so much nicer than commutative monoids, which often is taken for granted.
I am tempted to stump for the centrality of Galois theory in modern mathematics, but I feel that this subject is too close to my own research interests (e.g., I have worked on the Inverse Galois Problem) for me to do so in a truly sober manner. So I will just make a few brief (edit: nope, guess not!) remarks:
1) Certainly when I teach graduate level classes in number theory, arithmetic geometry or algebraic geometry, I do in practice expect my students to have seen Galois theory before. I try to cultivate an attitude of "Of course you're not going to know / remember all possible background material, and I am more than willing to field background questions and point to literature [including my own notes, if possible] which contains this material." In fact, I use a lot of background knowledge of field theory -- some of it that I know full well is not taught in most standard courses, some of it that I only thought about myself rather recently -- and judging from students' questions and solutions to problems, good old finite Galois theory is a relatively known subject, compared to say infinite Galois theory (e.g. the Krull topology) and things like inseparable field extensions, linear disjointness, transcendence bases....So I think it's worth remarking that Galois theory is more central, more applicable, and (fortunately) in practice better known than a lot of topics in pure field theory which are contained in a sufficiently thick standard graduate text.
2) In response to one of Harry Gindi's comments, and to paraphrase Siegbert Tarrasch: before graduate algebra, the gods have placed undergraduate algebra. A lot of people are talking about graduate algebra as a first introduction to things that I think should be first introduced in an undergraduate course. I took a year-long sequence in undergraduate algebra at the University of Chicago that certainly included a unit on Galois theory. This was the "honors" section, but I would guess that the non-honors section included some material on Galois theory as well. Moreover -- and here's where the "but you became a Galois theorist!" objection may hold some water -- there were plenty of things that were a tougher sell and more confusing to me as a 19 year old beginning algebra student than Galois theory: I found all the talk about modules to be somewhat abstruse and (oh, the callowness of youth) even somewhat boring.
3) I think that someone in any branch of pure mathematics for whom the phrase "Galois correspondence" means nothing is really missing out on something important. The Galois correspondence between subextensions and subgroups of a Galois extension is the most classical case and should be seen first, but a topologist / geometer needs to have a feel for the Galois correspondence between subgroups of the fundamental group and covering spaces, the algebraic geometer needs the Galois correspondence between Zariski-closed subsets and radical ideals, the model theorist needs the Galois correspondence between theories and classes of models, and so forth. This is a basic, recurrent piece of mathematical structure. Not doing all the gory detail of Galois theory is a reasonable option -- I agree that many people do not need to know the proofs, which are necessarily somewhat intricate -- but skipping it entirely feels like a big loss.
Best Answer
It is debatable that a physicist would use those very words, and if they did one would hope their meaning would be the same as for a mathematician, since it means that they are trying to speak the same language.
Having said, and coming from a Physics background, when I first learnt about filtered objects and associated graded objects, I immediately recognised the following examples from Physics. They all have to do with quantisation/classical limit in one way or another.
The Clifford algebra is filtered and its associated graded algebra is the exterior algebra. Under the "classical limit" map which takes the Clifford algebra to the exterior algebra, the first nonzero term in the graded commutator of two elements defines a Poisson structure on the exterior algebra. You can then view the Clifford algebra as the quantisation of this Poisson superalgebra. In Physics the exterior algebra is the "phase space" for free fermions and Clifford modules (=representations of the Clifford algebra) are Hilbert spaces for quantized fermions. Things get more interesting when the underlying vector space is infinite-dimensional, since not all Clifford modules are physically equivalent. (The relevant buzzword is Bogoliubov transformations; although you would not guess this from the wikipedia page.)
The algebra of differential operators on $\mathbb{R}^n$, say, is also filtered and the associated graded algebra is the algebra of functions on $T^*\mathbb{R}^n \cong \mathbb{R}^{2n}$ which are polynomial in the fiber coordinates (=the "momenta"). Again the first nonzero term in the commutator of two differential operators defines the standard Poisson bracket on $T^*\mathbb{R}^n$ and one can view the algebra of differential operators as a quantisation of this Poisson algebra. In Physics, this corresponds to quantising $n$ free bosons.
In both cases there is no unique section to the map taking a filtered algebra to the associated graded algebra, but one has to make a choice. There are number of more or less standard ones: Weyl ordering for the bosons, complete skewsymmetrisation for the fermions,...
By the way, this (and a lot more) is explained in the fantastic paper Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras by Kostant and Sternberg.
Edit (inspired by Mariano's answer)
Kontsevich's deformation quantization is not just of interest to physicists, but has a quantum field theoretical reformulation due to Cattaneo and Felder. It is basically the perturbative computation of the path integral of the Poisson sigma model. (This is analogous to how the perturbative evaluation of the path integral of Chern--Simons theory gives the Vassiliev invariants of (framed) knots.)
The picture that seems to be emerging is that indeed quantisation (be it deformation or path-integral or what have you) of a classical physical system gives rise to a filtered object, filtered by powers of $\hbar$.