I've always saw Dragonborn to be the fantasy race where you can display the Head of State, Head of Government, and Head of Military as 3 different people with important powers. The Dragonborn Republic would have an Elected Royal Clan, an Elected Council of Lawmakers, and an Elected Council of Generals. They'd have entire clan dedicated to war, art, or wealth but none of them would shun any of the three. An like you said, the main or first contact most other races would have of them would be another nation forced to pay tribute back to the republic after loss in a war.Necessarily, dragonborn must be "uncommon" because they're still pretty new, but I would put forward the following alternatives besides the classic fallen empire.
2. Martial Republic, likely a colonizing one. A fusion of Athens and Sparta: no slavery, but a sort of fusion of the idealized versions of both societies, with a dash of Sun Tzu, where the ideal soldier also studies poetry and art and philosophy. Unafraid to take land by force, but not warmongering just to warmonger. Possibly something of an upstart, or their homeland is far away and only their colonies are known. This emphasizes the long-term ambitions of the dragonborn, and gives a nod to the implicit higher degree of egalitarianism in their society since women are effectively equivalent to men (and don't need to spend several months pregnant.)
3. Nomadic Clans. Taking inspiration from the Mongols, Turks, Vikings, Sea People, and other highly-mobile, martially-inclined societies. Whether they sail, herd, follow a seasonal cycle, or some other motive, they don't often settle down and form civic societies--but they can still be a massively dangerous force to be reckoned with, especially if someone can unite them under a single common banner. As above, unafraid to take land by force, but usually too caught up in clan warfare to do so. This emphasizes more the stereotypical social structure ascribed to the dragonborn and their competitive streak.
And when the clan don't set aside differences to get along for long they become this. A bunch of competitive, self sufficient, and dargeerous class.. Embodying the power and selfishness of the dragons they come from.