Are neutrinos just different states of a single particle, in essence?

On both sides of your two equations, there are two distinct particles: The r.h.s. takes place in the "mass basis", where you identify particles $\lvert \nu_1\rangle$ and $\lvert \nu_2\rangle$. The l.h.s. lives in the "lepton basis", where you identify particles $\lvert \nu_\mu\rangle$ and $\lvert \nu_\mathrm{e}\rangle$.

These are just two different choices of basis for the space of neutrinos. The mass basis is natural when you look at stationary states, but the lepton basis is natural when you want to talk about interactions with other particles - it is the $\lvert \nu_\mu\rangle$ that participates in weak interactions involving muons, not one of the mass basis states.

On a more general level, a particle state is just another quantum state. And quantum states can always be expressed as superpositions of other states. I could define two "electrophotons" as the superpositions $\lvert \gamma\rangle \pm \lvert \mathrm{e}\rangle$ of a photon and an electron state and re-express all electron and photon states in terms of electrophotons. It just wouldn't be very useful.

Maybe it helps to see that this notion of re-evaluating our notions of the "best" choice of basis for a particle states is not exclusive to neutrinos: When we think about electrons and how the weak interaction only couples to states with a particular chirality, we also find that the notion of the "mass basis" electron can be expressed as the superposition of electron states with a definite chirality. It is the mass basis electron that we detect, but the chiral states that participate in the weak interaction (or not). For more on that particular case, see this answer of mine.

In the end, quantum states are simply not amenable to picturing them with our classical intuition. You're surprised that one can express a particle state as a superposition mostly because your ontology of "a particle" is intuitively classical - a little ball that sits in space. But that's not how quantum states work - state vectors can superpose in a way that classical states cannot.