Analogue systems are a powerful instrument to investigate and understand in a controlled setting many general-relativistic effects. Here, we focus on superradiant-triggered instabilities and quasinormal modes. We consider a compressible hydrodynamic vortex characterized by a polytropic equation of state, the polytropic hydrodynamic vortex, a purely circulating system with an ergoregion but no event horizon. We compute the quasinormal modes of this system numerically with different methods, finding excellent agreement between them. When the fluid velocity is larger than the speed of sound, an ergoregion appears in the effective spacetime, triggering an "ergoregion instability." We study the details of the instability for the polytropic vortex, and in particular find analytic expressions for the marginally stable configuration.