Actually... it's complicated. The Bank of England, for example, was nationalized only in 1946, and it remains technically a company which is owned by the state, not actually part of the government.
In the US, the Fed is... well, it's not a company, but it's also not not a company... or group of companies...
Yeah, the Bank of England even used to offer account services to employees after it was nationalized. Which would be scandalous if we found it in some dubious ex-Soviet republic today, albeit there maybe they're lending the Bank's President $10M with no real security and the Bank of England was offering employees mortgages and similar modest run-of-the-mill secured loans.
The UK owns a whole bunch of banks and entities that would need a banking license if they weren't owned by the government, of which only the Bank of England acts as a central bank, but there are also a bunch of commercial banks in (or at least operating in) the UK that are named after parts of the UK even though they're not owned by the government, including the Bank of Scotland.
To be fair, unlike Tether, the Bank of Scotland actually does hold actual cash worth the same as the notes they issue.
Also, although these banks have permission to issue notes (whereas if you went around issuing "bank notes" you'd likely get arrested) the notes aren't necessarily worth anything except in the sense that you can assume the Banks will give you Bank of England notes for them if it came to it since they're required to hold those.
They aren't legal tender (Scotland doesn't really bother having legal tender laws anyway) and retailers can choose not to accept them if they want. You won't have a problem exchanging them for tourist stuff in Edinburgh or buying a fish supper in Dundee, but good luck getting some random corner store owner down South in Cornwall or Essex to accept them - even though these notes are in some sense worth the same as Bank of England notes, there is no law requiring retailers to accept any notes and so they might just tell you to fuck off with your weird-looking money.
The decision-making body of the Federal Reserve System – the thing that sets monetary policy, among other things – is a government agency (the Board of Governors of the Federal Reserve.)
The rest of the system is, as you note, complicated, but the rest of the system doesn’t set monetary policy.
Actually... it's complicated. The Bank of England, for example, was nationalized only in 1946, and it remains technically a company which is owned by the state, not actually part of the government.
In the US, the Fed is... well, it's not a company, but it's also not not a company... or group of companies...
See https://en.wikipedia.org/wiki/Federal_Reserve_Bank