Hangouts received an update since this was posted. It now registers itself as a dialer with the system, and comes up as a choice when phone links are tapped in the browser app like Chrome. However, it's still not an option when you use built-in People/Contacts apps.
Hangouts Dialer app does not register itself with the system as an actual dialer. In fact, it's simply a shortcut to a particular tab within the main Hangouts app.
Right now if you want to make true VoIP calls, you have to open the Hangouts Dialer (or main Hangouts app) manually, and copy/paste the number. Perhaps such integration will come in a later version of these apps.