No, it obeys your commands, but can operate independently, as per the wording if the spell. Specifically, the wording says, "Your familiar acts independently of you, but it always obeys your commands." That means it is capable of acting on its own accord, s if it does not feel loyalty toward you, but rather obeys you begrudgingly (imps and quasits, I'm looking at you), then you had better be exact with your wording, or else that familiar will have a Wish, and that's not great for the master.
And to those of you who say, "Well I'll just make my wording perfectly exact," I just have to say good luck to you.