Socialism, by definition, is ownership of the means of production by the working class. Anarchism, by definition, is a system without hierarchy. If working people own their tools, and don’t rent them from a capitalist, what does that have to do at all with anarchism?
In reply to anarchism being capitalist, I disagree. Capital is a material representation of power. The people without it are forced (“choose” starvation) to work for the people who do, and there’s no system which makes sure that equality of opportunity is present. That is a system of hierarchy and therefore not anarchist. Anarchism is about more than the state.