First of all, I don't have Mountain Lion....I'm still running Snow Leopard. Hope I can still help you troubleshoot this.
Machinarium has a built-in Flash Player (10), and I'm thinking that *maybe* it is in some way not fully compatible with Mountain Lion??? So one thing you could try is to download Adobe's standalone Flash Player projector from this page:
http://www.adobe.com/support/flashplayer/downloads.htmlOnce it's downloaded, you'll have to perform a couple of additional steps:
1. Open the folder which contains Machinarium and Control-click on the Machinarium application.
2. Select Show Package Contents and then open Contents/Resources.
3. Locate the movie.swf file and move (or copy it) so that it sits right alongside the application for Machinarium.
4. Control-click on the movie.swf file and choose Open With, and from the list find Flash Player 11.5.
I'd be interested to see if that allows you to save the game.
Edit #1: Also, if you're using Safari as your browser, perhaps it's deleting your .sol files. Perhaps there is some change in Safari 6's preferences which is messing with your cookies; something checked or unchecked that was changed by default when you'd installed Mountain Lion. Check the Privacy tab of Safari Preferences. Is it set to block cookies?
Edit #2: Another thing...if you launch the Machinarium application and then click anywhere on the screen while pressing the Control key, you'll see a little menu pop up. Click on "Settings." Select the tab along the bottom which shows an icon of a Manila folder with a green arrow. Is the little slider set to 1 MB or more?