The popularity of taverns named the King's Head, the Queen's Head, and the Turk's Head (an evolution from Saracen's Head) in the 16th century would seem to make one very hopeful that the King's Arms referred to, well, the king's arms.

Alas.

"Arms" in English inn/alehouse names relates to coats of arms.

The use of heraldry in alehouse names and signs goes back to their origins in the Middle Ages! Barrie Cox did a study of English tavern names recorded between 1423-1426, and found that 80% of their names derived from some form of heraldic symbol. The "Lily" is the fleur-de-lis; there's the "Rose", "Swan" (VERY popular), "Hammer," &c.

The determining factor of medieval tavern naming conventions was the sign. Literacy rates even in late medieval cities were low--maybe 30-50%--so a sign with a concrete, easily recognizable picture was the way to go. And that's why you end up with taverns whose sign is technically a heraldic charge, but the actual name people used was the...well, the actual name of the thing. In 1423, the Moon tavern by London's Bishopsgate, not the Crescent was noted; in 1424, the Three Moons tavern at The Stocks.

(So in the late Middle Ages, you can already see the integration of numbers and colors into alehouse signs/names, although it's pretty rare still. Cox turned up a Green Gate, Red Cock, and White Cock, plus the Three Moons and Seven Stars. But that's out of 50 taverns total (and 41 unique names. Like I said, the Swan was VERY popular.) And he notes that except for the Three Moons, the earliest attested tavern with a triple charge sign/name is over a century later, in 1540.

Heraldry wasn't the only source for easily painted, easily interpreted, relatively easily recognized when someone was drunk inn names. For example, the Pewter Pot. Since religious iconography was also familiar to most people, relevant saints (St. Julian) are also represented in the record. Michael Camille notes that in France, St. Julian on a sign could either be the name of an individual tavern, or within a single city, the designation for generic "tavern" that would be accompanied by something specifying the name.