There is no rule that prevents these from being used together.
The dragonborn’s breath weapon states:
When you take the Attack action on your turn, you can replace one of your attacks with an exhalation of magical energy
The ascendant dragon monk’s Breath of the Dragon feature states:
When you take the Attack action on your turn, you can replace one of the attacks with an exhalation of draconic energy
Both features require taking the attack action, and no other prerequisites are mentioned. So if you take the attack action, you have met the requirements for using each breath weapon. Next, each weapon says you can replace one attack with the breath weapon. You can’t use both if you only have one available attack, but if you have two attacks available, there is simply nothing that prevents you from replacing both. The “Combining Game Effects” rule states:
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap.
Since these are different features with different names, they can happen at the same time, as long as you use your attack action and have two attacks available to replace.
While I don’t think it’s necessary for having a complete answer, I think it might be valuable to address a particular line of reasoning employed in Groody’s answer (deleted, 10k+ rep only), since it’s probably too long for a comment. Groody writes:
Assume you replace the first attack with the ability from the dragonborn race, and now you make a second attack: you want to now use the ability from being a monk. It says you can replace one of your attacks with exhaling energy when you take the Attack action. Well, you already did that, replacing another one would mean you replace two of your attacks. Does the ability allow you to replace two of your attacks? No. It says you can replace one attack when you take the Attack action, and you did already.
So as Groody states, begin by supposing we take the Attack action and use the Dragonborn’s racial breath weapon. At this time, we have replaced exactly zero attacks with the monk’s Breath of the Dragon feature. Next, we use Breath of the Dragon to replace our second attack. It is at this point that Groody claims we have now replaced two attacks, even though feature only says we may replace one. The issue with this is that each feature is telling you what you can do with that feature only, not telling you what you can or cannot do with other features. When a feature prevents or modifies the use of another feature, the feature will tell you that, or there will be a general rule governing their use. Since neither feature precludes the use of other features, and there is no general rule preventing us from using both, we can use both. Each feature has replaced exactly one attack, just like the feature tells us we can do.
Or to put it another way, after using the racial trait, we have replaced one attack with the racial trait. After using the class feature, we have replaced one attack with the racial trait.
