The system to log non EU citizens in and out is the passport, Maud. The Canadian's passport would need to contain a Schengen visa and a Schengen entry stamp. All the passport officer had to do was look at the date on the stamp. No database needed.
I can't speak for all countries but Greece does not record when people from the EU enter or leave the country, Scooby. Think back a few years. What happened when you went through passport control? You handed your passport to the officer he looked at you, looked at the passport and handed it back. How did he record your entry/exit? No such record exists. There is no such database. For people from countries where a visa is required the date of entry and exit is stamped in the passport, often on the same page as the visa. That is the only method of recording. The passport is the record. The current scanning is checking against various lists not recording entry/exit.
In the UK all attempts to record entry/exit data were abandoned by the Blair government in 1998. In 2015 the UK, under the coalition government, did start to make checks again and they claim that they are very effective but personally I have my doubts.
As far as I am aware when they scanned your passport they were checking
your name against various lists, e.g. see this press release
. You can also see the Directive here
. They were not recording
The only difference between using your Australian or your UK passport is that with the UK passport you don't, at the moment, need a Schengen Visa and the passport would not be stamped with a Schengen entry/exit stamp. With the Australian passport you would need a Schengen visa and the passport would be stamped with a Schengen entry/exit stamp. If you don't get a Schengen exit stamp within 90 days of getting the Schengen entry stamp then that fact would be recorded on the Schengen Information System when you do leave the country and you would have problems entering the Schengen zone in the future, even to a different country. Your name would show up on the Schengen Information System as someone who has previously overstayed a Schengen visa when your passport is scanned at the next entry. The stamps in the passport are the recording mechanism. The Schengen Information System is simply a way of disseminating information on the few who break the rules.
If/when the UK is fully out of the EU and any transition periods have expired, UK citizens will be treated like Australian citizens are now unless
they can prove right of residence with some sort of residence document. Initially that might be the existing Registration Certificate but is likely to change to some other form of residence document at some point.