Spanish usually uses ser for characteristics that are seen as more general or defining, and estar for states or conditions.
If you say:
that can sound more like My dog is being obedient right now or My dog is acting obedient at the moment.
So es is the most natural choice if you mean obedience as part of the dog's character.
Some Spanish adjectives change form for gender, and some do not.
But obediente is an adjective ending in -e, so it usually stays the same for both masculine and feminine singular:
It does change for number:
In this sentence, obediente is not directly attached to the noun. It comes after es because it is part of a predicate:
This is very normal in Spanish, just like in English:
If you put an adjective directly before or after a noun in Spanish, that is a different structure.