I, an American, only drop the name if it's going to be useful later, e.g. you're going to meet my girlfriend or son soon, or I'm about to launch into a tale in which it would be inconvenient to refer to people only by title. Or if we're on a level of friendship that socially dictates you can or should be made aware of my family members' names.
I can say (for me) it gets "weird" if you just keep referring to "my wife" for too long in a friendship. Sooner or later you'd have to give a name, or it sounds like you're trying to keep things impersonal--mixed signals. "The guy seems like he's my pal, but he keeps referring to his wife as 'my wife' and won't give a name." It feels like I'm not even worthy of knowing the names of people in your family. And maybe I'm not--maybe I'm just your coworker and that's where things end. But if we're hanging out, going on hikes and stuff, it'd be time to put a name to that title.
So though I don't do this (that I'm aware of?), I can imagine Americans putting the name out there early to encourage closeness, that you might think, "Oh, this person really wants to be friends since they already told me their son's name."
I know Americans have the related reputation of being hyper talkative and friendly with people they've never met and are unlikely to meet again. And that's just part of the culture, especially in smaller towns. We consider it to be welcoming, while foreigners might consider it intrusive and rude. I always tone it way down when I'm talking to non-Americans. But if I'm buying groceries, at the very minimum, the clerk and I are going to have a conversation about how our days are going, and probably what our later plans are.
And it would be a little weird to drop a family member's name at that point. But it would actually be acceptable to do so.