Gravitational Collapse: Kerr solution is a vacuum solution but not for any rotating body?

First of all, the Schwarzschild metric is the most general spherically symmetric vacuum solution to the Einstein field equations. A Schwarzschild black hole is a black hole that, having neither electric charge nor angular momentum, is described by the Schwarzschild metric.

The Kerr metric has a few "problems" and cannot be used to describe realistic stars apart from asymptotically far away. That is because realistic stars have:

• a) an interior. There are no acceptable interior solutions to the Kerr metric as they cannot satisfy required boundary conditions.
• b) close to their surface, their mass distribution and hence the surrounding spacetime have multiple moments in the multipole series expansions, which in principle can be independent. In the Kerr solution, the multipole terms are actually closely related to each other, so in order to match the star's $n^{text{th}}$ pole to that of the Kerr solution, it would require an unphysical gravitational collapse evolution which selectively radiates only certain poles.

A black hole has neither an interior nor it is undergoing gravitational collapse. So the Kerr metric may well describe the spacetime outside a rotating one. Hence the name Kerr black hole.