The federation is a post-scarcity socialist utopia. They don't even have money. Every single human being has ensured healthcare, housing, food, and education of their choice guaranteed from birth. Rise among ranks of the few hierarchical power structures is based on merit, performance, experience and training. I can't recall anything specific about the productive sectors that allow this to happen, but since they have access to virtually infinite amounts of energy and everything can be done by machines and matter replicators, there's no motive for hoarding means of production or wealth, so one would assume that most productive endeavors and enterprises are collectivists by default. Same with political institutions as hoarding power doesn't guarantee anything significant beyond what the average person already posses. They also have wide social openness, tolerance and acceptance as the most common sources of intolerance and bigotry (wealth, religion, power, prestige, etc.) have been regulated or removed. So there's no logical point on slaving, discriminating, oppressing or exploiting any particular class of people, some classes of people might not even exists, as there's no concept of poverty, nor race or sexual discrimination in the culture of the federation.

As a result people don't have to work, but most probably choose to involve themselves in some sort of productive activity as a form of hobby. Members of the Starfleet for example, aren't doing so for any particular material incentive. But do it because they think space exploration is neat, or because they seek glory and honor on the Starfleet mission, or because they really really like fusion cores.

They are as socialist as it comes.

Capitalism was eliminated on Earth by the New World Economy, which was likely a Dictatorship of the Proletariat as envisioned by Marx, Engels, Lenin, Roddenberry, etc. The Federation appears to be a classless, moneyless post-DotP society that still has one primary state apparatus (the Federation itself) that oversees many smaller state apparatuses (the Federation's many member-worlds). You'll notice a contradiction, though: If a state "is a system by which the ruling class maintains and perpetuates its dominance within the social formation... by subjugating the other class(es) within class society" then how can the Federation be a classless society? I propose two solutions:

1. Star Trek is fiction and fictional worlds are often incomplete and contradictory. Everything I've said about the New World Economy, the Federation, etc. should be taken with a grain of quadrotriticale.
2. No society has established a DotP, and there are certainly no examples of post-DotP societies. Marxism is a scientific and materialist worldview -- it has evolved since the 19th century and it will continue to evolve into the 23rd century and beyond.

EDIT: My answer is "Yes, but it's Advanced Sci-Fi Communism."

Yes, obviously. The Federation is a postscarcity socialist society

You can look up the definition and see if it applies. I'd argue it isn't a classless society. Especially with all the military ranks and hierarchies. And socialism is kind of a broad term. I'm pretty sure you can apply it to this case without starting a debate.

Neither, since they are moneyless and post scarcity. We honestly don't have a word for whatever they are.

There are capitalist elements. The Picard family still owns a farm and farm house and pass it down generations. There is still some concept of money being used by humans who are pursing payments for rare and stolen goods. Most of what we see around Starfleet is merit driven people working in starfleet out of self interest but the ships appear to be owned by Starfleet and they seem to have some democractic structure. Since most basic needs are met via replicators it seems they are post scarcity and trips to the doctors seem free but is not really socialism in the sense of people owning the means of production, there doesn't seem to be much of anything said about how these ships get built and the implication is its a lot of automation but there seem to be a lot of facilities on Earth with people in them like Starfleet academy and in bars. We have no idea how factories work in this world other than on other planets and people work in them.

I don't think its brilliantly clear, there are a mix of ideologies on display and what makes it hang together is the humans are all behaving well, which isn't very human nature like at all. People don't seem to own what they are working with in all cases but they do in some of the smaller settlements so its a bit of a mix dependent on circumstances.

inhaled very deeply about to talk about activitypub and techrights idealism then noticed this is star trek dot website

It's absolutely not a communist anarchy. There's a government, and it controls all those ships, science stations and mining operations. It doesn't look like URSS-ish communist either, as it's clearly democratic.

Besides, there exists some form of capitalism in it. It's just not very intense on the human worlds. And it's clearly socialist, as everybody is included on the society... So, my guess it's social-capitalist just like every advanced society today, just way richer than anything we know.