Yes, this system is called the
Visa Information System or short
VIS. This system stores travelers information, such as date, time, carrier of arival and departure. See here for more details.
Visa Information SystemIn case you need to prove the you left Shengen area, you should have asked the officer to stamp your passport. Since you do not have that, you can use your flight ticket as a supporting evidence. Your flight ticket has you name on it, is paired with your passport number and has the nature of official document.