Every country with some form of election has something analagous to a constitution, because that's what a constitution is: a description of how government is formed, performed, and powers entailed. Many countries without elections also have something similar. This being said, without a document specifically called 'the constitution', the phrase "that's against the constitution" doesn't get a lot of play.

Australia doesn't need it to be in the constitution that slavery is against the law. It's been illegal here since before the civil war (throughout the British Empire in the 1830s). We did have some slavery - 'blackbirding' - and it was illegal without having to be in a constitution.

There is absolutely nothing magical about a constitution, it's just that generally making changes to that particular piece of law is defined as being more difficult than other bits of law, given that it also describes how the whole thing works. In the US, if you can get your law attached to the constitution, it's harder to repeal since you need a greater majority to do so. That's all. There's nothing that makes rights more special because they're 'in the constitution'.

Ergo it doesn't have the same thing the US does.

The only thing it's lacking is a snappy one-page document.

... that can be referred to in front of a judge and supersedes everything else.