Why must the eigenstate of an operator correspond to a particle?

My question is, what is so special about the eigenstates of this operator and why do they correspond to particles?

In QM when identical particles are indistinguishable we require the quantum state describing the particles to remain the same under particle exchange. Since multiplying a state vector by a constant doesn't change the actual state, this is equivalent to requiring our multi-particle states to be eigenstates of the exchange operator.

About the latter part of the question, I'm not sure what you mean. What else would we be describing quantum systems of? Our quantum startes describe states of particles, and so we require our theories to match what we know about these particles and how they behave.

Actually it doesn't have to be from a mathematical perspective, but this only comes up in $3$ or more particles. Let's consider the case of $3$ particles as it's enough to illustrate the point.

For permutation group $S_3$ there is a 2-dimensional representation, meaning there are states $psi(x_1,x_2,x_3)$ and $phi(x_1,x_2,x_3)$ for which $P_{ij}psi(x_1,x_2,x_3)ne lambda psi(x_1,x_2,x_3)$, i.e. the permutation operator acting on a state does not necessarily return a multiple of that state. The condition $P_{ij}^2=mathbb{I}$ is still statisfied for this 2-dimensional representation even if $P_{ij}psi(x_1,x_2,x_3)$ will in general yield a combination of the $psi(x_1,x_2,x_3)$ and $phi(x_1,x_2,x_3)$ states. In the (canonical) Yamanouchi basis for instance, begin{align} P_{23}=frac{1}{2}begin{pmatrix} -1&sqrt{3} sqrt{3}&1end{pmatrix} end{align} is clearly not diagonal but does satisfy $(P_{23})^2=mathbb{I}$.

Kaplan in

Kaplan, I.G., 2013. The Pauli exclusion principle. Can it be proved?. Foundations of Physics, 43(10), pp.1233-1251.

argues that, if one allows such multidimensional representations, one obtains a fallacy based on the idea that "The physical picture in which adding one particle changes properties of all particles cannot correspond to a system of independent identical particles." In fact, the same author in

Kaplan, I.G., 1975. The exclusion principle and indistinguishability of identical particles in quantum mechanics. Soviet Physics Uspekhi, 18(12), p.988.

shows that the mean value of the one-particle operator $hat f (x_j)$ depends on the particle numbering if multidimensional representations of $S_n$ are allowed. In the example of that paper averages of $hat f(x_3)$ are not identical to those of $hat f(x_2)$ or $hat f(x_1)$.

What is special about the one-dimensional representations, i.e. the symmetric and antisymmetric representations, is precisely that all these average values are independent of particle labelling.

For completeness, the case of 2 particles is special because the irreducible representations of $S_2$ are both 1-dimensional, and in fact are precisely the symmetric and the antisymmetric representations, and nothing else. It is only for 3 or more particles that one can have multidimensional representations.

Thus, there is no reason to believe, based on permutation symmetry alone, that states must be eigenstates of an exchange operator. It seems that one must require additional physically relevant conditions to exclude states which are NOT eigenstates.

If you have the stomach for some more advanced mathematics, there's also a discussion of permutation symmetry requirements in

Hudson, R.L. and Moody, G.R., 1976. Locally normal symmetric states and an analogue of de Finetti's theorem. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 33(4), pp.343-351.