What is the number of polynomial factors of $a^n-b^n$?

As Lucian pointed out this follows immediately from the properties of cyclotomic polynomials. We have the factorization (into polynomials irreducible over $\Bbb{Q}$): $$ x^n-1=\prod_{d\mid n}\Phi_d(x). $$ Your observation follows from this because it yields the factorization: $$ a^n-b^n=b^n\left[\left(\frac ab\right)^n-1\right]=b^n\prod_{d\mid n}\Phi_d\left(\frac ab\right)= \prod_{d\mid n}b^{\deg \Phi_d}\Phi_d\left(\frac ab\right).$$ The multiplier $b^{\deg\Phi_d}$ exactly cancels the denominator created by plugging in $x=a/b$ to $\Phi_d(x)$.

The factors that you have found are exactly the polynomials $b^{\deg \Phi_d}\Phi_d(\frac ab)$ - one for each divisor of $n$.