Sharepoint - Is it possible to replace the personal site / MySite with a custom site template in on-premise SP 2013?

I do not think you can easily change the Site Definition that are used for creation of MySite personal sites.

The normal approach is to use Feature Stapling, to push in your own features "on top" of the Site definition. The features can have for example Feature Receivers that change specific things, and modules that adds Lists, pages, branding files).

Here is a good starting point: http://sharepointologic.blogspot.se/2013/04/branding-sharepoint-2013-my-sites-with.html where stapling is discussed in a branding mysite perspective.