This paper seems to suggest that the interior metric for a black hole in particular (a.k.a not a different kind of spherically symmetric non-rotating body) is just the exterior Schwarzschild metric but with the $t$ and $r$ coordinates switched. But this doesn’t seem right because the interior metric should really look more like the interior Schwarzschild metric. Am I missing something important here?

Here is the equation: $$\mathrm{d}s^2=-\left(\frac{2M}{t}-1\right)^{-1}\mathrm{d}t^2+\left(\frac{2M}{t}-1\right)\mathrm{d}r^2+t^2\mathrm{d}\Omega^2.$$