Schwarzschild metric: stars vs. black holes Background
The Schwarzschild metric can be used to describe the geometry of the vacuum spacetime outside a spherical massive object. For a star of radius $r$ (which is larger than the corresponding Schwarzschild radius $r_{S}$), we usually use the Schwarzschild metric to describe the spacetime geometry around the star for radial distances larger than $r$. (For reference see the top of page 287 in [1].) On the other hand, when we study the spacetime geometry around a black hole of radius $r_{S}$, we usually describe the spacetime geometry up until the interior singularity.
Question
My question is twofold. First, given a suitable mass density function could we extend the Schwarzschild metric into the interior of a star? Second, why is it that we describe the black hole until the interior singularity and not the star? Moreover, what is the difference in the metric between the center of a black hole and a center of a star?
Due to my lack of knowledge I understand that it is possible that my questions are ill posed. Hence, if that is the case, I would gladly welcome any clarifications.
References
[1] Bernard F. Schutz, A FIRST COURSE IN GENERAL RELATIVITY. Cambridge Univ. Pr., Cambridge, UK, 2009.
 A: 
Second, why is it that we describe the black hole until the interior singularity and not the star?

Because it can be shown that if the entire mass of an object is inside its Schwarzschild radius then it will collapse to a singularity according to classical general relativity. See Tolman–Oppenheimer–Volkoff limit.
The Schwarzschild metric is only valid in the vacuum around a spherically symmetrical object. For a classical black hole, the Schwarzschild metric will work for any point apart from the singularity which contains the entire mass. Your confusion probably arose because you thought that there is mass everywhere inside the event horizon. But classically everything apart from the singularity is vacuum.
After including quantum gravity effects, things will become complex and are not well understood. Quantum gravity usually won't allow exact singularities. In string theory Fuzzball and in loop quantum gravity Planck star are some alternatives to exact singularities.
A: You may want to look at the interior Schwarzschild solution, which describes a the metric inside a spherically symmetric incompressible mass of constant density and with zero pressure at its surface.
It is (in geometric coordinates) $$ ds^2 =\frac{1}{4}\left(3\sqrt{1-(r_s/r_g)} - \sqrt{1-r^2r_s/r_g^3}\right)^2 dt^2 - \frac{dr^2}{1-\frac{r^2r_s}{r_g^3}} - r^2(d\theta^2 +\sin^2\theta d\phi^2) $$
where $r_s=2GM$ and $r_g$ the r-coordinate for the surface.
It is worth noting that the pressure at the centre becomes infinite for $r_s=(8/9)r_g$ (the Buchdahl limit), and this is likely the definite limit for any material. In practice stars are much less compression resistant.
Note that the metric at the centre $r=0$ is totally well defined and friendly as long as the pressure is finite. Overall, for a given time-slice it is basically a (hyper)spherical cap smoothly joining the exterior (vacuum) solution.
Now, doing this kind of analysis for a more complex density or pressure field is way more involved. And doing it for a dynamical situation like a collapsing star is far harder: there are no known real analytic solutions to that problem for realistic matter fields.
A: Like you said, the Schwarzschild metric applies to a static, spherically symmetric, vacuum spacetime. The Schwarzschild metric applies up to the singularity of the black hole because the region interior to the event horizon is a vacuum: $G_{\mu\nu}=T_{\mu\nu}=0$. The interior of a body such as a star is obviously not a vacuum so the Schwarzschild metric does not apply. Instead you would have to solve the field equations for an appropriate energy-momentum tensor, subject to the condition that the solution of course matches the Schwarzschild metric at the surface. For example, a simple model could be that of a perfect fluid of density $\rho$ and pressure $P$: $$T_{\mu\nu}=\rho u_\mu u_\nu+P(g_{\mu\nu}+u_\mu u_\nu),$$ where $u^\mu$ points in the direction of the timelike Killing vector field. You can find a detailed treatment of this case in section 6.2 of Wald (1984).
