I am only talking about matrices for finite number of states.
By the existence of unique equilibrium distribution, this surely means there can only be one of such eigenvector (i.e. the eigenvalue 1 has geometric multiplicity of 1)
Is it possible such a matrix to have eigenvalue 1 repeated?
For any $n\times n$ stochastic matrix $P$, it is true that $$\left|\lambda_i\right| \le 1,\quad i = 1, \ldots, n.$$ Also, we can easily prove that $\lambda = 1$ is an eigenvalue of matrix $P$. Combining these facts, it holds that the spectral radius of $P$ is $\rho(P) = 1$.
Due to irreducibility, we may apply the Perron - Frobenius theorem to the stochastic matrix $P$ and we get that $\lambda = 1$ is a simple eigenvalue.
Thus, $\lambda = 1$ cannot be repeated.