This is only a pet peeve because I’m a huge dog lover, but no, Fido is not a “member of household” under State Department regulations. Despite the fact that pets have an enormous impact on employee moral, especially for officers posted in less than ideal posts, the State Department does not make any special accommodations for your pets.
Housing Example: Fido chews through your government-provided Drexel Heritage dining room table; you are on the hook for making the table whole again either through repair or replacement.
Shipping Example: Fido needs to get to China with you? Get ready for a possible 30-day quarantine in Beijing Customs once doggy or kitty arrives in country. Doggy has to get to Dubai with you? If he’s snub-nosed, no airline will fly him to Dubai (Middle Eastern airlines have lost a few snub-nosed dogs due to respiratory issues and none will take these types of dogs again). If he’s not snub-nosed but over 20 pounds, doggy will have to fly by cargo and not in the cabin with you. Keep in mind most air cargo holds are not climate controlled. As you can see, the list of logistical challenges in shipping a pet could be very, very long.
Behavior at Post Example: Somehow your pet gets out of your house and causes an incident that effects the community, such as doggy bites the guard to your house or community enclave or kitty is decimating the local endangered bird population. In a case like this, post management (which includes the Ambassador) could require you to remove your pets immediately at your own cost.
Lastly, in many places, veterinary care and attitudes towards pets is not to US standards and customs. In leasted developed countries be ready for emergency contingency planning and possible mistreatment of your pets if you leave it unattended in public places.
In sum, the official policy of the State Department to traveling pets is that they are a privilege but not an entitlement. That being said, there are many sympathetic colleagues (such as me) who can help iron out some of the wrinkles in safely getting your pets to your overseas post and settling in safely and happily.