At first I thought that you might be
finite simple groups (you a cyclic group of prime order, learned about in your first year, then maybe $A_5$, from the class of alternating group and a group of Lie type (dynamics), etc). Could also have been just sporadic simple groups, although those don't come up during your freshman year.
but there were too many clues that I couldn't explain.
Then I thought you could be
There are still many parts that I doubt about, but let me try, first the global story:
I think you are some kind of a fermion, probably an electron, which may be the only elementary particle you study during your freshman year. Your first friends in the second year were some more fermions, but by the end of the sophomore year you made friends that gave rise to some dynamics: those were some gauge bosons, like the photon, and maybe weak gauge bosons: in quantum field theory, fermions interact through the exchange of bosons, so without bosons there are no dynamics. The classmates of your first 2 friends defined your whole group of friends, even though several are from other classes. That is because the gauge bosons arise purely from the fermion dynamics when you impose local gauge invariance.
In junior year, more friends, but a simplification anyhow. Could that be the addition of quarks, which revealed that hadrons and mesons were not actually elementary but had substructure, and instead of having scores of hadrons and mesons, you could do with a handful of quarks?
Looking in the mirror, you might see your antiparticle. It might also be you superpartner. Not sure in which sense your group of friends is exceptional though.
Classes could be fermion generations or particle types (fermions, vector bosons, scalar bosons) or also fermion types (leptons or quarks). To make it work in this story, different ones of these could serve.
To fill in the details, there are different stories possible, none of them much more convincing than any other, so I'm not so sure.
Some clues that I have no idea how to interpret:
Not sure about the d12 and d20, they seem to be dice shaped like a dodecahedron or an icosahedron, but since both have the same symmetry group, it seems unnecessary to change one for the other in the context of elementary particles.
No idea what happened in senior year...
Let me make a wild guess, I hope that I'm at least somewhat on the right track:
The classes are three generations of leptons, then quarks (let's restrict them to one class) and bosons: 5 classes.
You are the electron, your first friends where the muon and the tau-lepton, then the photon, two quarks and your new friends from senior year, but let's not count them because I don't know how to interpret the clue. You would have at least 5 friends, but possibly also quarks of different generations and additional gauge bosons. The Higgs doesn't seem to enter, and neutrino's not explicitly. Most of these are different from their antiparticle, so if you befriended those as well, there are many more. The quarks also have a color charge, so if we count those as different friends, there would be many more again.