As an initial remark, the Schrodinger equation can be solved exactly for a variety of potential, not just “hydrogen-like” atoms - the cases of the harmonic potential, the Morse potential, or Poschl-Teller potential immediately come to mind but there are multiple other ones as well that don’t have entries on Wikipedia.
Quantization of angular momentum follows exactly from the commutation relation of the angular momentum operators $\hat L_x,\hat L_y$ and $\hat L_z$. These commutation relations do not depend on the number of particles in the system.
The simplest example would be two spin-1/2 particles, made famous by various versions of Bell’s theorem. The total spin (or angular momentum) of this 2-particle system remains quantized and it can only take the values $0$ or $1$ depending on how the state is prepared.
Another example is the nuclear $su(3)$ model - not so used in chemistry but still quite useful for multi-particle nuclei. The 3-dimensional harmonic oscillator can be solved using the $su(3)$ Lie algebra, and the angular momentum operators are in this algebra. It is also “easy” to construct multiparticle $su(3)$ state using Lie algebraic techniques (just an extension of Clebsch-Gordan technology for angular momentum). For these multiparticle states, angular momenta of the individual constituents are combined in the usual way and remain quantized.
More generally, when the potential is not central, angular momentum will not be conserved for individual single particle states and so states will not necessarily be eigenstate of angular momentum. An example of this is the Nilsson model for deformed nuclei. This does not mean that basis states with good angular momentum quantum number cannot be used to start the calculations, just that the final states will be a linear combination of those single particle basis states and that $\Delta \ell\ne 0$.
Of course since the total Hamiltonian must be a rotation scalar, the final eigenstates of $H$ can be chosen to have good angular momentum. Angular momentum remains quantized since the total angular momentum operator still satisfy the usual commutation relations.