Neutral anions in salts and strong acids are similar - is it a coincidence?
Though couched in different terms, the first part this question is essentially the same as this other one:

Does a strong acid&#39;s conjugate base act as Br&#248;nsted-Lowry base?

No, this is not a coincidence.  The conjugate base of a strong acid will always be a weak base.  The 'neutral salt anions' of your professor's nomenclature are just weak inorganic Brönsted bases.



The reason why $\ce{SO4^2-}$ doesn't appear in this list of 'neutral salt anions' is because it actually can noticeably affect the $\mathbf{pH}$ of a solution when added as a salt.  $\ce{H2SO4}$ is often treated as though it's a diprotic strong acid, but in reality only the first deprotonation is strongly acidic.  This can be seen by examining its acid dissociation constants; from Wikipedia:


  $\mathrm pK_{\mathrm a1} = -3$
  $\mathrm pK_{\mathrm a2} = 1.99$


This $\mathrm pK_{\mathrm a2}$ is about the same as the first $\mathrm pK_{\mathrm a1}$ of $\ce{H3PO4}$ $($$2.148$$)$, and is actually somewhat less acidic than nitroacetic acid $($$1.68$$)$.

So, in some cases sulfate may act more or less as a 'neutral salt anion', when the $\mathrm{pH}$ is sufficiently high that it will all stay doubly deprotonated, but in others it can definitely influence the solution $\mathrm{pH}$. I've actually used sulfate/bisulfate before as an acidic buffer for various projects at work—works great!