I do not understand this mania mortgage brokers appear to have for not giving you their last names. They're willing to do it if you ask twice, but keep saying something like "Oh, you don't need my last name."

I mean, no, I don't. But we are in a business relationship, and the current social climate suggests that there is nothing horrendously unprofessional about relaying your last name. I might not use it if you introduced yourself by your first name in your office, but I would expect either a nameplate or a business card that actually had your family name to be somewhere in evidence.

First names only are common[1], but casual. They're what you get when the friend you're out with introduces you to someone they know during the chance encounter at the coffee shop. It's *weird* not to know the last name of anyone you're contemplating paying notable quantities of money to[2]. I'm guessing it's pretty clearly an attempt to foster at least some of the effects of a close casual relationship, and I'm guessing they may actually get in trouble if they don't do it.

(But a small part of my mind is still crying out "Yeeeeek, people! Back off, already!")

Small ramblings.
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[1] Much as it occasionally annoys me.
[2] In a legitimate enterprise, at any rate. Big Louie and Jimmy the Bat can carry right on.