I can always count on readers to offer explanations. James Begent informs me that "an esquire is an old term for the rank below a knight. In modern use it's more of a term of respect, since the meaning doesn't apply any more. People like the bank will use it to be obsequious."
There you go.
Mr. Begent, esq. also says on his bloggy sort of site: "I believe there is no meaning or purpose in anything. All effort is futile. All desire is empty. Buddha was nearly right but when you die you rot in the ground. God is Santa Claus for adults. Humans are monkeys. Life was an accident. I like burning things, especially hackers. I like complexity and abstraction as they distract me from the futility of human existence".
Hmmm. Cheerful. I tend to agree (except about burning hackers, but this is probably some computer jargon witticism that I am not acquainted with).
Update: More input from yet another reader - "We lawyers in the U.S. often have Esq. after our names." Aha!