For the most part, sure. But that's the parent's point, I think. Central banks are not companies, they are part of the public financial infrastructure of a nation (or, in the EU case, group of nations).
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...
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.
But they can't issue banknotes, unlike BoS and RBoS.
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 rest of the system is, as you note, complicated, but the rest of the system doesn’t set monetary policy.