Are there atoms with different numbers of protons, but the same number of electrons, with different orbital occupations?
Yes, the number of protons in the nucleus does affect how the electrons are filled into orbitals. There are a number of real-world examples that usually stem from transition metals since they do not only have a rather rich oxidation state chemistry but also two energy levels (3d and 4s) which are rather close by each other.

To give you a concrete example, take $\ce{Fe^2+}$ and $\ce{Cr^{\pm 0}}$. They both have the same number of electrons. However, chromium’s ground state electronic configuration is given as $\mathrm{3d^5 4s^1}$, while iron(II) typically has $\mathrm{3d^6 4s^0}$.

The general explanation is that the higher nuclear charge in the iron nucleus attracts the electrons better than the lower nuclear charge in chromium. This difference causes the orbitals to contract. The d-orbitals which are closer to the nucleus, are contracted more and thus experience more stabilisation than the s-orbitals of the next shell.