We study a single-server priority queue with a finite number of classes, in which the arrivals follow a fractional Poisson process of index $α\in (0,1]$ and the service completions are triggered by an independent fractional Poisson process of index $β\in (0,1]$. Each of the customers arriving is assigned at random to one of the priority classes. This assignment is independent of the rest of the system and follows a fixed probability distribution. Using a time-change representation of a fractional Poisson process, we first give a multinomial thinning decomposition: the total number of arrivals in each class are independent standard Poisson processes of appropriate intensities, time-changed by a common independent random clock that is the inverse of an $α$-stable subordinator. This yields a process-level law of large numbers and a functional central limit theorem for the process of arrivals. For the queueing system itself, we identify process-level scaling limits for the cumulative and individual queue lengths of the classes. We also prove that the queue gets empty infinitely often when $α\le β$, which does include the critical case $α= β$. A final example shows how the model can be extended to a continuum of classes.