At the time of European invasion *cough* settlement, the guesstimate is 200. The guesstimate is based on poor data, since many were wiped out so quickly, and on Lexicostatistics
— because we don’t have enough data to make a good linguistic assessment of what counts as a different language otherwise.
We could of course ask the indigenous Australians themselves how many languages there were. Their answer will be closer to 600, which was the number of distinct tribes — because like people everywhere else, their notion of what counted as a distinct language was based on identity politics rather than mutual intelligibility. (I see that Wikipedia is now using the identity politics answer rather than the white linguists’ answer: Australian Aboriginal languages ).
That’s a distinct answer from how many languages are alive today, being passed on to new speakers, and not revived from written sources after going extinct. That answer is closer to a dozen or so.
How similar are they? Tasmanian is very poorly attested (likely nine distinct languages jumbled up in a concentration camp), but we are pretty sure they have no detectable relation to the mainland languages. The bottom 2/3 of the mainland belong to the Pama–Nyungan languages, meaning they have discernable similarities. Given that pana means “man” in one end of the family and nyunga means “man” in the other end, they aren’t that similar. The guesstimate is that they are of a similar time-depth as Indo-European, 5000 years.
The top 1/3 of the mainland are heterogeneous, and are not particularly close to each other; it doesn’t seem to have been definitively proven that they are related to Pama-Nyungan, either.