The Dirac delta function can be defined as $$\delta(x)=\frac{1}{2\pi}\int_{-\infty}^\infty e^{itx}dt$$ From this we see that the dirac function has units of $x^{-1}$.
This doesn't follow. The Dirac delta function produces a unitless value, typically used as a factor for some other term to zero out that term's value for most values of $x$.
Sources I can find for the momentum eigenvector ignore the units of the delta function without even mentioning.
They're not assigning units because the Dirac delta function itself lacks them.