Hospitals are run and funded by each state in Australia. I think a decade ago the federal government started to run a hospital in Tasmania, but they pulled out of it because it was ridiculous. There are also numerous private hospitals. GPs and specialists are largely privately operated. Comparing Medicare to a single nationalised hospital system is not accurate — it's more like a national health insurance system with membership fees paid by the tax system.

Obviously having a single purchaser of so many services allows them a large degree of control, but it isn't even a single payer system, since many health services require extra funds (even a routine doctor's visit requires out of pocket costs for many people, like $30 — that isn't cheap) although that's a recent change. There's also private health insurance and extra fees you can pay for nice service (like a room of your own).