In the US, the "head of government" is the president, not the Speaker of the House. The president does not require the support of the majority of the legislature which is why the POTUS can be from one party while Congress is controlled by the opposing party. So, not essentially the same.
In our parliamentary system, the "head of government" is the King of England, not the Prime Minister. The king's representative in-country is our Governor General. The King holds a similar position in the government that the US president does. Although their powers and duties are significantly different, of course. Generally, the British monarch doesn't give a fig about Australia and doesn't get involved at all, and the local government is left to run stuff. Also, the monarch is not a member of any political party.
While they are elected by the people, the name "Prime Minister" designates that office as being a minister
to the monarch, to whom they make the oath of office (unlike the USA, where the oath is to uphold the constitution). The PM is the notional leader of the government of the day and is afforded considerable authority, privilege, and levers of power to get things done. So much so that they have often considered themselves unassailably "head of government" and behave as such.
But while they are generally left alone to run and represent the country,
any ministers including the PM can still be directly dismissed by the monarch at any time (they "serve at His/Her Majesty's pleasure"). This happened for us in November 1975, suddenly highlighting this hierarchy of power. Rude wake-up call.
Granted, this is a complicated colonialist hangover, and a patently ridiculous arrangement. I'm one of those Australian Republicans who would prefer an Australian Head of State and much better government design.