I received a login notification via email or Kakao Talk message although I have not logged in.

When you log into Kakao Game, Kakao Story, Kakao Talk, etc. using Kakao Account email address and password, login history is sent to you via Kakao Talk or your Kakao Account email address.

If it is an act you do not recognize, you can protect your Kakao Account by clicking "Change Password" at the bottom of the notification.