This is a OS issue and you did not state what OS you are running. You have to set the default program to open a file within the operating system.
If you are on Mac OS, then right click on the icon, you should get an "Open With..." drop down, from here you can set default application (see screenshot)
I am running Windows 7 Professional and even when I choose "Open With" I have to browse to FileMaker Pro 12 to choose it.
It still opens with Adobe Acrobat and defaults back to Adobe Acrobat as the "Open With" program.
This does not happen on all computers. I have computers that will open in FileMaker Pro and a couple that try open with Adobe Acrobat. The computers are all built with the same image.
In Windows 7 you click, right click on a database that should be associated with 12, then select "Open-With", select "Choose Default Program", and then select Filemaker Pro 12.
If you skip the step "Choose Default Program" then it will only open in that program that one time.
I am selecting "Choose Default Program".
Choose the default program to open with on the database not the shortcut.
Delete the shortcut, then recreate it Right Click your desktop, New, Shortcut, Browse to the database and select it, click next, name your shortcut, click finish.
If you still have problems I would scan for a virus, because there would be another program changing the default program to open the database.