Some novels seem to say that in village life (or within a hutong) poorer children often had no personal given name. Some characters in realistic seeming novels are known only by family name and birth order and have no other name. But I have only seen this in novels and I wonder if it really happened.
In Europe even into the 20th century some people had no family name but were known only as say "Nathan's son John" where Nathan was literally the father and not a family name. On the other hand, in places where childhood mortality was high, many children would not be named until they reached a safe age of 10 or 12 years. Such a child might be called "The oldest Jones girl" or "the middle Jones boy" but I do not think it was ever so organized as "Jones one, Jones two...."
Many good sources discuss highly developed Chinese customs of having personal names and public names, and changing these over a lifetime, in well-off families. I have not found good historical sources for naming practices among farmers or the poor.
Were some people in the past actually known just by names like 张三?