The people discussed in Ragtime are a mixture of fictional characters created by Doctorow, such as the family and Coalhouse, and the author's representations of real historical figures like Houdini and Evelyn Nesbit. People from both of these groups regularly interact with members of the opposite group, though not in extremely significant ways, such that the conversations that Doctorow claims they had aren't truly disprovable historically, that they could have plausibly happened.
The first significant example we see of this type of intersection of real and fictional characters is when Houdini literally crashes into the family's life. In most other books, this would seem ridiculous, as the protagonists just meeting Houdini out of nowhere is really unrealistic. However, I didn't view it as Houdini crashing into the house of the protagonists, whose story we are hearing, rather the story is being told about the family because they are the family whose house Houdini crashed into. This kind of thing occurs throughout Ragtime, as some people in the family have close connections with Coalhouse, Evelyn Nesbit, and Robert Peary, but I still wouldn't say that this makes the family special. They're just a normal, generally remarkable family, who happen to have interactions with some more influential and famous people.
I believe that this theme is what Doctorow is getting at with his choice not to name the members of the family. While the historical figures in the story obviously need to have their names, Doctorow is clearly trying to say something by naming some of his created characters, such as Coalhouse, while leaving others with simply titles. This shows that Mother, Father, and the boy aren't significant to the story on their own; they could be replaced with any other family including a mother, a father, and a child. It's said at one point that Father feels like he is spectating the relationship between Coalhouse and Sarah develop, that he's somewhat distant from it and is just watching it take place. It's the named characters in the novel; the Coalhouses, the Evelyns, the Emmas, who truly move the plot along and shape it. The unnamed family, on the other hand (besides Younger Brother, to an extent) don't truly have a huge effect on the plot themselves; they are just involved with the people that do and for that reason are able to witness it.