you are correct. The Schrödinger equation for the hydrogen atoms in spherical coordinates demonstrates separability which allows you to separate the radial and angular coordinates. The radial term, which is most interesting due to the 1/r potential is typically a Laguerre polynomial. The angular term is 'free' from any potential is typically a spherical harmonic.

The spherical harmonics in general are typically derived as part of the solution to the Laplace equation in spherical coordinates. A bit of a semantic point (though perhaps the distinction is important) though, since the Laplace equation's angular dependence is identical to that of the Schrödinger equation for the hydrogen atom.