Whether a dragon's breath weapon is considered magical is specifically addressed in the Sage Advice Compendium:
Is the breath weapon of a dragon magical? [...] First, Cold Breath isn’t a magic item. Second, its description mentions no spell. Third, it’s not a spell attack. Fourth, the word “magical” appears nowhere in its description. Our conclusion: Cold Breath is not considered a magical game effect, even though we know that dragons are amazing, supernatural beings.
The same is true of dragonborn characters from the PHB: "You can use your action to exhale destructive energy. Your draconic ancestry determines the size, shape. and damage type of the exhalation." This does not mention being magical, and so the same strictures from the S.A.C. presumably apply.
However, Fizban's Treasury of Dragons changed this. The chromatic, metallic, and gem dragonborn species presented in that book say this:
Breath Weapon. When you take the Attack action on your turn, you can replace one of your attacks with an exhalation of magical energy in a 30-foot line that is 5 feet wide.
This does claim to be magical. Now, this seems like a deliberate change, so quite possibly the Fizban's team explicitly decided that if you're using the Fizban's dragonborn, the breath weapon is now susceptible to being suppressed inside an antimagic field or similar, and that this is an actual change from the previous version. This feels like a straight-up deficit for a dragonborn character, which isn't ideal, but equally the Fizban's dragonborn has some straight-up improvements (breath weapon use governed by your proficiency bonus rather than short rests, for example) so perhaps this balances out?