Dragon Shape says:
You can take on the form of some of the world’s most fearsome creatures. Add the forms listed in dragon form to your wild shape list. Whenever you’re polymorphed into another form using wild shape, you gain resistance 5 to your choice of acid, cold, electricity, fire, or poison.
If we look at Dragon Form it says:
Breath Weapon (arcane, evocation) The shape, damage, and damage type of your breath weapon depend on your specific dragon form (see below). A creature in the area attempts a basic save against your spell DC. This is a Reflex save unless stated otherwise in the special ability description for your specific dragon form. Once activated, your breath weapon can't be used again for 1d4 rounds. Your breath weapon has the trait corresponding to the type of damage it deals.
The breath weapon is not listed as a spell, instead the breath weapon is just an ability you can use, similar to how a Mu Spore's Grasping Tendrils is just an ability. The breath weapon having the Arcane trait, does make it magical in nature though.
Widen Spell states:
You manipulate the energy of your spell, causing it to spread out and affect a wider area. If the next action you use is to Cast a Spell that has an area of a burst, cone, or line and does not have a duration, increase the area of that spell. Add 5 feet to the radius of a burst that normally has a radius of at least 10 feet (a burst with a smaller radius is not affected). Add 5 feet to the length of a cone or line that is normally 15 feet long or smaller, and add 10 feet to the length of a larger cone or line.
Since Widen Spell only applies to the Cast a Spell action, and the breath weapon is its own action, Widen Spell does not apply.