Australia & America
Legislative Branch Similarities
Both the Australian and American legislatures have a Bicameral Parliament/Congress, where the House of Representative as the "people's House" and the Senate as the "State's House".
Both Houses are directly elected by the people in both of the countries.
Strong bicameralism is present (i.e. each house has co-equal powers) in both Australia's and America's legislatures.
Both lower houses in the two countries have the sole power to initiate Money Bills or Appropriations.
Executive Branch Similarities
Both country's executive has a Cabinet, which can act as an advisory council to the PM/President.
Australia and America's Executive branch plays a large role in the decision over the budget. In America the House of Reps & the Executive decide the provisions within the annual budget. This is similar to Australia, where the Executive within the House of Reps (established by Westminster conventions) decides on the aspects within the budget.
Another similarity is that within both countries, the executive is the one that approves the bills passed by both Houses. In Australia, theoretically the Governor General approves and may withhold approval from bills - however this has not yet occurred. In America, the President may refuse to sign into order proposed bills - vetoing - which can be corrected via a supermajority veto.
Judicial Branch Similarities
The Australian and American judiciaries are almost identical, with both being comprised of federal courts and judges, as well as possessing a constitutional court.