Cats were apparently extinct in the Fallout universe (despite some supposedly being used to create Centaur mutants when exposed to the FEV), until Fallout 4 included cats. There are cats living in some settlements, a cat that came from a vault, and actually dead cats found in the Institute.
So one of a few things might be true:
Cats survived in New England, but not in former Washington DC, the Mojave Desert, or anywhere near the Californian west coast. (Though somehow, a cat is found on the Pyrdwen. An exotic pet, by Brotherhood of Steel standards?)
A secret cat vault was opened somewhere in New England, releasing domestic cats into the wasteland.
Test subject cats somehow managed to escape the Institute.
All cats in the Commonwealth are synth cats specifically populated by the Institute.