In practice, it did not, and the advent of other class delimiters such as sex, race, ethnicity, are later grafts usually but not entirely ascribed to the Frankfurt School. Here once again only "false consciousness" leads someone to consider himself a Frenchmen or a Canadian instead of primarily a man or woman, black or white, Italian or Jewish. But Marx was correct about what this would do to his theory of classes - because people have memberships in multiple such classes (and nationality is just as much a one as the others) the solidarity is diluted and relations between the classes become muddied - as they are in practice. Whatever this is, Marxism it is not.
Marxian theory turned out to be quaint and rather old-fashioned over only a little time. By 1860, when he began Capital, the predictions he and Engels had made in the 1848 Manifesto concerning the proletariat and capitalism in general - increasing illiteracy, falling profits, etc - were demonstrably not the case. But Marxian techniques, that is, the agitation of class rivalry and the employment of amoral manipulation to break down the structure of the existing society - these turned out to be, and remain, very successful at gathering political power. It is these that so blur the distinction between classic Marxism and the neo-Fascist model in currency with the Left these days. Most classic Marxists are fine with that because they still hold that any breakdown of existing society must ineluctably result in a more socialist future - Marxian historiography "proves" the inevitable progress of history. But breakdown is possible, building a new socialist society, not so much. It doesn't stop the True Believers from trying but it cautions anyone sane against the results.
Thanks for the very interesting note.