What is Marxism? It is historical materialism as a means to progressively change the world. Why does Marxism as a political movement centre on the working class? Because of its position firstly as in independent class with its own aims, second because of its essential role in creating wealth, third because it owns no property in the means of production when it assumes power it has no other group to oppress or exploit, therefore workers power can be used as a means to escape to a classless society.
When we define working class we must insist on wage labour being the key determinant and not things like education or dress or personal consumption. Useful though this may be it is paper abstraction . There is no one who is simply a wage worker. People who insist we are working class and working class only get nowhere. Such an idea can pander to various prejudices, i.e. by suggesting women workers, ethnic or religious minority workers, LGBT workers etc must get over their petty oppression and deal with 'real' working class issues.
What defines wage labour? I remember thinking through why socialists advocate class analysis rather than patriarchy theory. The answer being the former relates to economics while the latter relates to ideology and of course men as a whole do not exploit women as a whole (exploit being the key word, rather than oppress). The former is more immediately economic than the latter but we would be abandoning historical materialism to insist there is an ideology without a basis. Wage labour is in part determined by women's oppression, as it is by racism, homophobia or any other systematic prejudice you can name.
So long as we're clear exploitation is the foundation of capitalism there is nothing wrong with trying to incorporate aspects patriarchy theory into Marxism.
