Mew begins with the French verb muer, meaning to molt. This derives from the Latin mutare, meaning to change. Thus, mew shares origins with mutate.
So the story goes like this: a mew was a cage to confine hawks when they molted, but eventually became all hawk housing.
A falconry bird is usually housed in a mews.Wikipedia
From specialized hawk/falcon housing, mews was extended to all hunting animals (horses, hounds, hawks), or more generally stables. The most famous mews is the Royal Mews at Charing Cross. The important characteristic of this mews turned out not to be the stables per se, but the courtyard that the stables surrounded.
With the demise of horses for transportation, stable were converted to housing for people with the desirable characteristic that the housing opened onto a pedestrian alley or courtyard, instead of a street. These a mews is an upscale development around a pedestrian space, as in the Washington Mews in New York City.
So from molting hawks, mews has become a small neighborhood around a pedestrian-only alley or courtyard, from housing for royal animals to the royals themselves.
Burrich had quarters over the stables, not far from the mews.Assassin's Apprentice by Robin Hobb