Consider 2 non-interacting Fermions with spin $s\ne 0$ trapped in a 1D harmonic potential with ground state $\left|\phi_0\right\rangle$. Due the Pauli exclusion principle I would expect that only 1 electron could occupy the ground state. Now if we add the spin of the Fermions as a quantum number (the spins not interacting with the potential) suddenly we could have 2 different states for the Fermions: $\left|\phi_0\right\rangle\otimes\left|s_1\right\rangle$ and $\left|\phi_0\right\rangle\otimes\left|s_2\right\rangle$ with $s_1 \ne s_2$.

So do the Fermions both have ground state-energy or do they need to have different energy? What if I add additional quantum numbers (e.g. without any physical interpretation, just constants)?

Consider 2 non-interacting Fermions with spin $s\ne 0$ trapped in a 1D harmonic potential with ground state $\left|\phi_0\right\rangle$. Due the Pauli exclusion principle I would expect that only 1 electron could occupy the ground state. Now if we add the spin of the Fermions as a quantum number (the spins not interacting with the potential) suddenly we could have 2 different states for the Fermions: $\left|\phi_0\right\rangle\otimes\left|s_1\right\rangle$ and $\left|\phi_0\right\rangle\otimes\left|s_2\right\rangle$ with $s_1 \ne s_2$.

So do the Fermions both have ground state-energy or do they need to have different energy? What if I add additional quantum numbers (e.g. some internal degrees of freedom with no effect on the Hamiltonian beside increasing the size of Hilbert space)?