Is differentiating particle and quasiparticle meaningless?

You may want to look for a discussion in Hakens "Quantum field theory of solids". In principle, one can quantize any field, by expanding it in its eigenmodes and imposing the commutation relations. The quantization procedures are thus the same in the QFT and the solid state, where one has such effective particles as "free" electrons, holes, phonons, magnons, polaritons, excitons, cooper pairs, etc. Some of these are often though of as rather simple two-particle combinations (excitons or cooper pairs), but they are really just the poles of the many-body Green's functions, and therefore "particles". Thus, from the mathematical point of view, they are not different from the QFT particles.

One calls these particle effective to distinguish them from the real particles of the QFT. In some sense it follows from our thinking of them being somehow secondary excitations of a solid state, composed of real particles. But since the QFT particles transform into each other, and can themselves be represented in terms of "more elementary" particles, this distinction is more semantic than logical. One could however argue, that the quasiparicles in the solid state are constrained by symmetries rather different from those in QFT, which mostly have to do with a crystal structure. For example, thinking of electrons and holes in solid state as electrons and positrons of the Dirac equation quickly runs into many obstacles: there is no Lorentz symmetry, the gap and the mass are unrelated, there are multiple hole varieties, etc.

TL;DR
Mathematical descriptions of quasiparticles and real particles are identical. Physically, one cannot meaningfully distinguish them on the basis of what emerges from what, but their properties are rather different.