In a capitalist society, class is a relationship to economic power, not a social identity. Here's why:
1. The goal for social identities is social equality. The goal for the working class is a fair distribution of wealth for everyone, regardless of social identity.
2. Most members of a social identity want to keep their identity. Few members of the working class want to keep working for capitalists.
3. In a capitalist society, anyone with capital is a capitalist; anyone without it is working class. You cannot change your race or gender, but you can change your economic class overnight if your luck is very good or very bad.