ZDF is not state-owned. It's an independent non-profit institution financed in part by the non-government TV tax.
It's like saying "independent courts" aren't part of "the government" because the courts are "independent" and because elected politicians don't have the official authority to tell judges how cases should be decided. Semantics. The courts are a branch of government.
That's an important part of their independence.
Also, it's not really a tax, it's a levy or fee.
In Germany, the right of those broadcasters to collect that fee is derived directly from the constitution, just like the right to own property is.
An ostensibly independent organization that has been granted a special right to collect a tax is an example of the former, not the latter.