You can make the claim, but it isn't "proper" in any way, shape or form.
We can't be both.
We were a "Confederation of States" with a constitution at one point as well. The key common denominator was a constitution.
The Continental Congress adopted the Articles of Confederation, the first constitution of the United States, on November 15, 1777.
You choose to argue, not deductively from the Constitution that you presumably wish to identify yourself with as I in fact have argued but instead you resort to argument by analogy, a weak form of disputation second only to argument by ad hominem. A vehicle that is a truck is not a vehicle that is a car-that enlightens us in some respect?
Perhaps worse, you are just factually wrong. You certainly can have both, in fact, in a decent political system you must have both.
Our Constitution provides for democracy by representation. People's democracies or People's republics, such as the Soviet Union under Stalin, claim to have a Constitution but have in fact ruthless rule by autocrat. They had a fascinating written Constitution but they were not a constitutional republic anymore than they were a democratic republic or a representative democracy. A Constitution without legitimacy born of democracy, is a meaningless scrap of parchment.
A democracy without a Constitution is mob rule. History is replete with examples aping the French Revolution of democracy without a Constitution and the rule of law. Equally history teaches us of so-called constitutional republics exploiting a written Constitution to cover lawless suppression of democracy.
God chose to assemble a unique collection of selfless patriots who contrived a Constitution that is a waypoint between the autocracy of a monarchy and the terror of mob rule. They succeeded in giving us both a constitutional republic and a representative democratic republic, they gave us both, thank God.