Such specs should be considered as "per channel."

Stereo has 2 channels. Each speaker, according to your specs, needs 80 watts and, therefore, each channel of the receiver should be capable of 80 watts (at least).

Not generally. Your ears will likely tell you when you are pushing too much and guide you to the volume control. There is somewhat greater concern of having too little power and driving the amp into clipping before achieving satisfactory listening levels.