I'm not sure how to choose in this poll because it always depends on the kind of narrator you're using, as Jenny-Toons already said, and the kind of story, as well.
When characters meet for the first time, though, why wouldn't they introduce themselves to each other? It would feel oddly unnatural if people just go about an adventure or something without even asking one another how they want to be called. So when characters meet for the first time, I usually just have them mention their names and that's it. In-character, of course.
In my dark fantasy (not online here yet), the protag is super distant, but meets a young girl just bubbling with energy – naturally, she reaches out to him, with name and all, while he........ doesn't. xD But the readers know his name, since it's third person limited anyway.
Maybe think about what would be natural in real interactions, and then apply that to your writing? When do people usually learn the names of others? Are they brought up by friends? Do they have to ask? Is the new person someone who's widely known by a lot of people so the name (and their status, e.g. a politician or something like that) is just something your protag knows and drops in mid-thought?
There are so many ways to show characters' names to the reader that I can't really think of all of them right now, but try not to artificially hold them back just because you don't think people asking each other's names feels natural. It is very much natural in real interactions, so I don't see a reason to avoid it in fiction. You don't even have to do it in dialog-form! Just mention somewhere that they introduced each other if it's not that important and then start using their names.
It's always more awkward to read about "the blue-eyed man" or "the blonde girl" for more than one time. Like... I know what they look like the first time these things are mentioned, no need to awkwardly spread it out. xD