It’s certainly government. The US Congress created it as an agency of the US government, just as it did every other government agency; and the Congress can destroy it. The Congress did not provide for direct political control—elections (and electoral politics) can’t change the Federal Reserve’s decisions (or decision-makers) directly.

There’s indirect control: the elected administration selects people to nominate as governors, who then go on to make the decisions. But they have to choose people who the (elected) Senate will confirm, and there’s not too much the executive can do if they don’t like the decisions their appointed governors go on to make during their 14-year terms.

The question at hand at the moment is whether the executive can unilaterally usurp the agency’s Congressionally-mandated structure and use its power directly.

It’s kind of like how the various Civil Service acts sheltered career civil servants from the constant changes in political winds, so that we can have career professionals in government instead of using public paychecks purely as a prize for the buddies of whoever won the last election. The present administration resents that independence too.