What could be causing the black spots in my orange cat's mouth?
The black spots in the mouth of an orange cat are simple freckle-like nevi, a benign pigmented skin condition.
It's actually quite common and can be found on humans. Although it looks a bit like freckles, it's not the same thing. 
The difference between simple freckle-like nevus and freckles:
Simple freckle-like nevus: There may be an increase in the number of melanocytes, and the shape will not change due to sunlight exposure.
Freckles: The number of melanocytes does not change, but the color will darken due to sunlight exposure.
The causes of simple freckle-like nevi are as follows:
Simple freckle-like nevi are formed due to increased melanocyte activity leading to pigment deposition. They are related to genetics and are most common in cats carrying the orange gene (orange cats, calico cats, tortoiseshell cats), but can also be seen in cream and silver cats.