In Snap-ins, When the offline chat option is enabled. The user initiate the chat and submit. So where does the case/chat is routed that is who is the case owner?
It goes to Web to Case provided you've properly set it up and configured it properly. To learn more, I recommend you go through the Web Chat Basics Module on Trailhead. The Create Pre-Chat and Offline Support Forms section of that Module walks you through how to setup and configure offline support.
As to who will own the case, that will depend on how you configure Web to Case's Assignment Rules. Depending on Assignment Rules, it could go to either a Queue or a specific user.
If it's the same thing I'm experiencing, this is a known issue where it does not follow the assignment rules. More here: https://success.salesforce.com/issues_view?id=a1p3A000000JWhVQAW