game of thrones - Is all the nobility in Westeros permanently drunk?

What's "drunk" ?

Drinking alcohol in limited amounts doesn't neccessarily make you "drunk" as per classic definition of that words. A typical adult male doesn't become drunk by drinking a pint of beer with a meal.

It will make you "drunk" as per definition of drunk driving in most western countries, since even low levels of blood alcohol content increase risk of car accidents. However, given the much lower alcohol levels of common drinks of the time, some two cups of "ale or wine with every meal, even breakfast" would result in a BAC of 0.05 percent or below, which has some relaxation and talkativeness effect (and would make you illegible to drive nowadays), but is strictly different from the common meaning of the word "drunk" which would imply the behavior of someone with 0.15+ BAC.

There are cultures nowadays where it's common to have beer or wine with every meal, possibly including breakfast, and it doesn't make anyone "permanently drunk" by any reasonable understanding of these words.

Not only nobility

As per the other answers, you'd expect the same habits for almost everyone else, including women and children. A nine year old daughter of a random tailor would also most likely drink small beer or watered-down wine at lunch and dinner.