I don't know if I'm using the App Store version – I've had a license for a very long time and it just updates itself.

Out of curiosity I launched the Cookie app itself from my Applications folder and I'm now able to view the browser-specific settings. This has also caused Cookie to start appearing again when I select "Show Cookie" from the task bar menu – before it had not been showing up at all. Somehow every website I've visited for a given timeframe was selected and marked as a favorite - so effectively Cookie wasn't removing any data at all.