Unlikely any official age.
The common theory is that during medieval times water sources were not particularly clean and therefore beer, wine, and mead were common beverages during meal times.
However, they were not like the alcoholic beverages of today. Typically children were served what is known as a small beer which has a lower alcohol content than normal beers such as ale. The same is true of mead, which had widely varying degrees of alcohol by volume (ABV) but was easily watered down. It is even possible that the Wort (the mixture of sugar before fermenting) might have been used as a beverage which while still sanitary had no alcohol.
Although, children would obviously be more susceptible to getting drunk on even smaller amounts of alcohol and parents would have to regulate how much their children consumed. This well explains Sansa's quote about being allowed only one or two cups while she was still in the 12-14 age range.
They dined on trout fresh from the river, and Sansa drank more wine than she had ever drunk before. "My father only lets us have one cup, and only at feasts," she confessed to her prince.
"My betrothed can drink as much as she wants," Joffrey said, refilling her cup.
They went more slowly after they had eaten. Joffrey sang for her as they rode, his voice high and sweet and pure. Sansa was a little dizzy from the wine.
A Game of Thrones - Sansa I
Arya is allowed to have beer in small quantities even at age 9.
Her father sometimes let them have a cup of beer, she remembered. Sansa used to make a face at the taste and say that wine was ever so much finer, but Arya had liked it well enough.
A Clash of Kings - Arya II
I would take the solider asking Arya about if she old enough to drink, not as bad writing but perhaps as, "Are you old enough to handle the alcohol.", much like she was by a common-folk;
"Sharna's husband, and an orphan boy they took in. They won't harm you. There's ale, if you think you're old enough. Fresh bread and maybe a bit of meat." Tom glanced toward the cottage. "And whatever you stole from Old Pate's garden besides."
A Storm of Swords - Arya II
Of course their are your standard options; milk and iced milk are common in Westeros, and water was drunk just not as the main source of hydration as it today.