Yes, I know we have to register (and registration is compulsory) but we don't have to register with or for a political party - a previous poster, perhaps p'dog, said he registered as a Democrat. We don't register for a particular party - or I'd be in strife. I usually vote for whoever's pissed me off least in the previous 12 months or so! (I'll have to rethink the state scene, now that just about everyone got thrown out yesterday! Politics! BaH!)
I struggle with the entire US political system.
The difference is due to your following the Westminster System and we don't.
You elect a Parliment and the PM is whomever can command leadership of the majority of that Parliment. In your case, you vote and the chips fall where they may. Whichever Party controls the Parliment controls the PMs office.
In our case we have no Parliment. We do have a similar House of Represenatives but the composition of that body doesn't determine who the head of state will be. The majority party of the House of Reps. and the party of the President can be (and often are) different.
In our case, the Political Parties decide amongst themselves (through Primary elections) who will be their appointed (or annoited if you prefer!
) candidate for President. The rules for running the Primary elections vary from state to state but the traditional general rule of thumb is that by registering with a party you have the ability to participate in the decision making. (This has changed some in recent years with some states going to "Open Primaries" and such.). If you weren't registered with a Party then you weren't allowed to help decide who the Party's candidate would be (you could still vote in the egeneral elections however).
The practical effect is that the power/authority that your MPs hold to decide who will be the head of government is held by all of the individual party members here.