Actually, in quite a few ways - although it is the Australian Monarchy within Australia (it's a technical distinction, because the British Monarch and the Australian Monarch are physically the same person - but legally and constitutionally they are separate and that distinction is constitutionally important).
The powers of the Crown are still very real in Australia. The Queen (normally through her appointed representative resident in Australia, the Governor General (currently General Sir Peter Cosgrove - former Chief of the Australian Defence Force)) has the power to dismiss from office a Prime Minister, or an entire government that seeks to act illegally or unconstitutionally - and this isn't just some theoretical power. It happened in 1975, getting rid of the most socialist Prime Minister we've ever had before he could cause very serious damage (he was not dismissed because of his politics - but because of massive financial mismanagement that included unlawful activities by members of the Cabinet and a refusal to accept that he could no longer govern effectively and refusal to call an election in those circumstances). Members of the Defence Forces and the Police Forces take their oaths to the Queen as symbolic of the fact that there is a power above Parliament and in the event of a government acting outside the law and constitution, this means they have a duty to oppose that government. None of this is trivial.
And a significant part of the republican movement in Australia comes from socialists who don't like the idea that they can be restrained from acting outside the law and constitution. Not all republicans are motivated by that - but a significant number are.