Familie is a singular noun, just as family is in English. You wouldn't say "The family have hunger" because the subject is singular. You would use haben for something like "Die Familien haben Hunger" ("The families have hunger"). Maybe the feminine die is what makes it seem even more plural?
Hat isn't "is". When we say hat Hunger or hat Durst, it means "is hungry" or "is thirsty". "The family has hunger (or thirst) is not commonly used. At least we don't say it in English. If you want to say "is hungry" or "is thirsty" in German, you would say ist hungrig or ist durstig. (Note that they aren't capitalized, because they aren't nouns like "hunger" or "thirst"). So there are 2 ways of saying "The family is hungry": Die Familie hat Hunger or Dir Familie ist hungrig. So getting back to your original question, hat is "has" not "is", so you are correct. Hope this helps. :)