TLDR: it is not socialist as it leaves the possibility for a capitalist system to exist in the community. That is, unless every single person refuses to enter into a contract where they work without the means of production (an untenable assumption unless you assume everyone will inherently and always make altruistic choices).

An anarchist market socialism is inherently impossible, due to the fact that it requires every single individual from making a very natural type of agreement, and such complete individual consent without community control is impossible.

In an anarchist market system, one is free to own whatever one is able without an overarching community or government to override an individual's decision. This may be "socialist" insofar as people collaborate to determine how they act in the market.

However, in order to be truly an anarchist market economy, one must be able to own what one wants. Under that type of system, it is an immediate and natural consequence that someone would, say, pay for someone to help them with a tool they own or manage something they own. Unless the community forces them to stop (which would no longer be anarchist), they have entered under an agreement where the owner of the means of production is not the worker, which is inherently not socialistic.

Further, the fact that this can occur makes the system inherently incompatible with socialism. In one instance you can choose to deny the core tenet of socialism, that the community can override an individual's market decision to work without ownership of the means of production. Alternatively, you can choose to deny a core tenet of anarchist market economies, that a collective cannot override an individual's market decisions.

The problem with your argument is that you are only considering particular individuals decisions, rather than the rules and principles which allow those conditions to be general. A system that has a company owned by workers is not in a socialist system if another company is owned by a capitalist. What makes it socialism (as I describe before) is the community overriding the ability to even have a company where the workers do not own the means of production.

The only way this is viable is if every single member of the society refuses to participate in that sort of agreement, completely without collective overruling. However, this is inherently untenable as an ideology for any large scales, as that sort of active consent needs to occur completely voluntarily for every individual. If we were capable of making such unanimous individual decisions, people wouldn't do bad things to eachother anyways.