The Green Dragon's breath is described as:

Poison Breath (Recharge 5-6). The dragon exhales poisonous gas in a 90-foot cone. Each creature in that area must make a DC 22 Constitution saving throw, taking 77 (22d6) poison damage on a failed save, or half as much damage on a successful one.

It is "poisonous gas" the Dwarf gets advantage on the save.

Poison. Venomous stings and the toxic gas of a green dragon’s breath deal poison damage.

It is "poison damage" the Dwarf takes half damage (effectively 1/4 if saved 1/2 if not).

With respect to your comments on protection from poison, I think you are splitting hairs: it is poison so advantage is given on the save; the fact that the effect is to do poison damage rather than impose the poisoned condition is irrelevant to the save.

That said, the DM is the sole authority on rulings so, check with your DM.

Both of these are DM calls.

So I'm wondering if a character could get drunk while wearing a PoPAP? There's no "drunk" status condition in D&D 5e, but drunkenness is caused because you've consumed enough alcohol to poison yourself, however slightly.

The problem here is that (outside of things that deal Poison damage or grant the Poisoned condition) there isn't any really great definition of poison. "The dosage makes the poison," and even water can cause intoxication and death in sufficient doses.

We usually bunch things that kill us with a "small" dose into the poison category, and things that kill us with a "large" dose into the not-poison category, but there's a lot of leeway in the middle.

Things that are immune to poison being immune to alcohol is a common trope. Alcohol isn't necessary for us to live, and has a variety of negative consequences. Thus, poison.

On the other hand, the effects of alcohol are pretty mild in typical dosages. It's unlikely to kill you, and may not even have a significant impact on you if you consume it slowly. Thus, not poison.

Which way the magic judges, is up to your DM.

Like for instance, if someone put spider venom in their food, would they know the food was poisoned when they ate it?

I think the general rule for spells applies well enough here. For spells, it works like this:

Unless a spell has a perceptible effect, a creature might not know it was targeted by a spell at all. An effect like crackling lightning is obvious, but a more subtle effect, such as an attempt to read a creature's thoughts, typically goes unnoticed, unless a spell says otherwise.

So, does the spider venom have a specific flavor, smell, or texture? The character can still discern that. If not, then the character doesn't get a "your amulet of poison has protected you" warning.

Of course, your DM could just as easily rule that you can feel the magic working. That there's a tingle in your stomach as the poison is attacked, or that the periapt glows, or similar.