If HDMI sound is a different card(like in my case i have ATI HDMI and Creative Xfi) then YES you can assign on to just a different program as long as the software supports it.
This issue comes that a program has to be able to choose what sound card it wants to use.
In this setup, i have SPDIF(digital, just like HDMI, in fact part of the HDMI standard) playing all computer sounds, but Winamp playing over analog speakers.
As said, your software or at least some of it has to be able to direct sound to what output it wants to use.
So lets say you are using netflix(uses default audio device), and can not change that, do your other programs have the ability to remap(many games support this, but i am not sure if a web browser does.)?