Depends on the anarchist. Many would focus on seeking the absence of involuntary power hierarchies. A manager who distributes work and does performance evaluations isn’t intrinsically a problem, it’s when people doing the work can’t say “no, they’re a terrible manager and they’re gone”, or you can’t walk away from the job without risking your well-being.

Anarchists and communists/socialists have a lot of overlap. There’s also overlap with libertarians, except libertarians often focus on coercion from the government and don’t give much regard to economic coercion. An anarchist will often not see much difference between “do this or I hit you” and “do this or starve”: they both are coercive power hierarchies.
Some anarchists are more focused on removing sources of coercion. Others are more focused on creating relief from it. The “tear it down” crowd are more visible, but you see anarchists in the mutual aid and community organization crowds as well.

Anarchists recognize class as a social construct rather than a biological imperative or a free market condition. As a result, they will often make a point of transgressing or undermining the pageantry that class-centric organizations cling to.

Its not that they think “no classes” will be a result so much as they think “explicitly defying class” is a political act.

It’s actually right in the name. Anarchy from an-arkhos means “without ruler”. They think hierarchies are illegitimate per se.

They define anarchy differently from the common definition. Anarchists believe in creating community organizations to serve the needs of society, but they refrain from calling it a state because they believe a state requires a monopoly on the acceptable use of violence.

They don’t think that we should just dissolve society and let everyone fend for themselves to eliminate class, unless they’re an edgy teenager.

Oh boy…

That’s functionally the difference between Anarchism, a fundamentally Individualist and Idealist ideology, and Marxism, which is fundamentally Collectivist and Scientific.

A Marxist political society will also tend towards Classlessness and Statelessness, though in the case of Marxism both of these are not goals but an inevitable result of a society dominated by the Proletariat according to Marxist theory.