As far as Schengen rules are concerned, it doesn't really matter: you can always prove your right to be in the Schengen area by showing your Slovak passport, or indeed by using any other means to prove that you are a citizen of Slovakia, regardless of what document you used to enter. What's more, there is no Schengen-wide database of Schengen entries and exits wherein you will have a "dangling" entry record on your AUS passport.
Because of the absence of a Schengen-wide system, any country's own system can only be relatively useless, since it's possible to enter and exit the area through different countries. It's therefore unlikely, but possible, that whatever country you entered through has a record of your entry as an Australian, but they won't be able to come after you for that. The most likely consequence of that is having trouble entering Schengen in the future using an Australian passport. Any such problems can be resolved by showing yourself to be a Slovak citizen, and avoided entirely by not presenting yourself as an Australian citizen.
The only thing I don't know about is whether you can get in trouble under Slovak law for having used your Australian passport. I know that there are a handful of European countries that wouldn't care much about this, and I don't know any that do care about it, with the possible exception of the Netherlands, but I certainly don't know how it is in Slovakia.
The fact that you have different names in your passports could cause some trouble in some contexts (such as with airline tickets), but it's not likely to do so in this context. The passports both identify you with a photograph and biographical information, and probably also with biometric information. You have a legitimate reason why the names are different. Officials in some countries might take exception to the different names, but it's not likely to happen in the EU.
Immigration restrictions and rights, such as the EU right of free movement, are generally applied to each person, not to the document on which they entered. Some countries may allow or deny access to certain rights and privileges based on the document you used to enter the country, but this is neither the case in the EU, nor in Australia.
If you haven't been able to find official information that you must present yourself as a Slovak citizen to Slovak authorities, it of course doesn't mean that such a requirement doesn't exist. But it does imply strongly that no such requirement is being actively enforced.
Finally, you may want to consider showing both your passports when you leave. This will resolve any questions about names not matching with airline tickets, and it will allow Slovak officers to do whatever they think necessary with any records they might have of your Australian passport, while also proving your right to enter, reside in, and exit Slovakia and the Schengen area, and the European Union.