There are five requirements that a government must pass before it can be considered a government: Authority, Law, Sanctions, Sovereignty and Succession.
As far as authority goes, the family is right up there. Mom and Dad are the authority in the family. All problems and major decisions are run by them.
Concerning law, the family is a prime example: the parents lay down the law and the kids obey. Simple, except that most of the time, especially with teens, they disobey. It’s a big problem.
Sanctions. This one’s a biggie. The parents inflict punishments on the children if they disobey the law of the family. Without question, this is one of the staples in family government.
Succession is taken for granted: although there are exceptions, most kids will likely continue the same regime which governed the household they grew up in.
The last requirement—sovereignty. Mom and Dad are the sovereigns of the household. No question about that.
Conclusion: The family is definitely a form of government. It has successfully passed all the requirements of any government for any faction of society. The family is one of the most important forms of government in any society. It is also one of the most bureaucratic.