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.

The absence of evidence is not the evidence of absence.

In this case, at least, the above is true. Since the wording for wild shape does not explicitly say conditions are removed/dispelled, and disease is a condition, that condition is not dispelled upon shifting back. It can then be surmised that status affects such as poisoned, blinded, etc, are also not removed upon shifting to and from wildshape, it can be easily concluded that those effects still persist in the original druid's form.

This is not to say a disease is a standard condition like poisoned, blinded, etc. But it is still a condition that persists on the player until the effect wears off, or is cured.

Diseases specify what creatures are affected by them. For example, Cackle Fever:

This disease targets humanoids, although gnomes are strangely immune.

Or Sight Rot:

This painful infection causes bleeding from the eyes and eventually blinds the victim. A beast or humanoid that drinks water tainted by sight rot must... etc.

After this point, the real question becomes does a mundane disease that affects a specific animal affect the druid in their humanoid form?

That goes into some science that is more theory than actual rules. So there's no easy way to say that a disease will or will not affect the druid in a different form. However, it could be said that the disease remains dormant in an unaffected or immune host, but when the host becomes a viable creature the disease persists. This is up to your DM at this point.

• If the druid is considered their race in terms of biological susceptibility in wildshape or not, then the disease would always persist.
• If the druid is not considered their race in terms of biological susceptibility in wildshape, then the disease would not persist, or would be quickly cleansed from an unsupported host.

One last observation from this can be taken from the text of Wild Shape:

You retain the benefit of any features from your class, race, or other source and can use them if the new form is physically capable of doing so.

Of course, biological susceptibility to a disease wouldn't exactly be considered a "benefit" but you could rule this to mean that if you are a Gnome you will, for the purposes of disease viability, be able to be infected as if you were a Gnome, no matter what beast you were taking the shape of.