In British English this is not the case, "Government" is use more like "Administration" is used in the US.
What do Brits use to refer to British courts, Parliament and executive considered as a whole?
Those are the three branches of the state. We consider government to be the executive only. The judiciary does not govern.
Ah. Americans use "state" to mean one of the 50 provinces of the US, leaving the word unavailable for what Brits use it for.
What about the state department?
That is our term for what the Brits call the Foreign Office.
Yes. I think the GP's point was that this shows how US usage also, still, to some extent reflects that of the rest of the world: Your State Department is called that because it is the organ that deals with other sovereign states.
Pointless word semantics aside, the point here is that they are branches of one thing; if you piss off one branch, strings can be pulled so that another branch treats you severely in court. This is plausible.