It is essentially the same system of government... three branches that check each other... call them branches our not, it is irrelevant.
No it's not
It is a presidential form of government - much favored by tin-pot third world dictatorships like Syria, Libya, Egypt, Zimbabwe, Uganda, Nigeria, Argentina (pretty much all of South America actually)
The judiciary should NEVER be regarded as a branch of government. It should be seen as 100% independent from it
Even here, in your Parliamentary Oasis, Judges (Judicial) check the Legislative and the Prime Minister (Executive) tries to lead but can be blocked by Parliament (Legislative) and people are represented by Ministers that vote (Republic Style) on our behalf, etc.
Oasis? - you mean the democratic system that
every other Western democratic country uses (other than the USA - with a note that France tried to cobble together a hybrid system of a parliamentary government AND with a president with certain powers)
In the UK judges do interpret the law and a challenge can be made to government policy. But the judiciary is very jealous of its independence from government
The Prime Minister is but part of the executive. He/she is the "first amongst equals" as the saying goes and it is the
cabinet that decides policy
The government must control the confidence of the legislature at all times and will fall if it loses a no confidence vote
The Prime Minister must retain the confidence of his/her party and again will fall if they lose it (as happened last year when PM Theresa May had to resign and Boris Johnson too over)
It is a flexible system with much better check and balances than the USA has with its presidential system
In the UK, there is a healthy disrespect towards government and the PM in particular is not treated with some kind of holy reverence that Americans have for their president
I think the best illustration of this was in 1980 when the Olympics were held in Moscow
Following the Soviet invasion of Afghanistan in 1979, President Reagan and PM Margaret Thatcher both ordered their respective Olympic teams to stat away.
No American athlete went to the Olympics
Almost the entire British team gave a collection F-You to Thatcher and went
The British government cannot and wouldn't dare put a country off limits (except in wartime) but you try going to Cuba and see how much jail time you get when you come back to the USA.