if you look at the end of the link you posted for the downloadable zip you will see it says ... moodle-20.hotpot-module.zip. The file is named like this because it contains the HotPot module for Moodle 2. It does not contain the QuizPort module.
As Glenys pointed out (thanks Glenys!), the QuizPort module does not run on Moodle 2.x. For the time-being you will have to make do with the HotPot module for Moodle 2.x, or continue using the QuizPort module on your Moodle 1.x site.
The database tables you saw being installed on your Moodle 2.x site were probably for the HotPot module, since it was the HotPot module that you downloaded. Probably you can see "HotPot" now in your Settings -> Site administration -> Plugins -> Activity modules, yes?