Obviously, there were models in Greece and Rome, and more modern republics such as Venice, as well as the beginning of a parliamentary system in Britain. How different was the US system from the others existing at the time, and how was it perceived in Europe?
The Super Powers of Europe had a rather different governmental structure that what we are familiar with today. Of all of them, England was the closest match to what we would consider a democracy, even if they couldn't vote for their king. The power of the Royalty of Great Britain was slowly being phased out by the Parliament and the biggest gripe of the Americans was that they were being taxed without being represented in England, something they felt was a fundamental right as British subjects.
To it's credit, it was still fairly autocratic in comparison with the Engand we know and love today.
As for the other powers of Europe, autocratic would be putting it mildly. France's monarchy wielded absolute power over it's subjects and it was considered the norm of Europe. The idea of the great unwashed masses determining who would lead them was shocking to the nobility and royalty of Europe. They had ruled for so long under the Divine Right granted to them by the church that it would be unthinkable.
While there were some glimmers of democracy across Europe, such as in Sweden (where a king was subject to the will of the people unless he had a really good idea or was in war) and Poland (which had a proto-democracy with the Polish-Lithuanian Commonwealth), there were places like Russia where serfdom was still enforced.
It wasn't until America was colonized that we begin seeing the beginnings of real democracy, usually modeled after what the English were doing at home. The Benelux countries has their own revolutionaries who were espousing the freedoms of democracy and managed to re-frame the debate about democracy equating to anarchy.
Then the French Revolution hit, and changed everything.
It terrified those in power and empowered the weak. It sent shock-waves through Europe and altered the political landscape forever.
In simple terms, it was not unique.
Very few forms of government are truly radical, as political thought and systems never emerge as if from a vacuum. The American republic did not emerge from nowhere. The influence of the political concepts evinced by theorists such as Thomas Hobbes, Jean-Jacques Rousseau and most especially John Locke is evident in the Declaration of Independence and Constitution. Locke et al were in turn influenced by the thoughts of others who wrote before them.
Quentin Skinner and J Pocock both argue that the revolutionary political writings of the enlightenment had its roots in the Renaissance humanists, most especially in the work of Niccolo Machiavelli.
Ernst Cassirer was a German Philosopher of the 20th century, in his posthumous work Myth of the State he wrote that medieval scholasticism was held together by a 'common center of thought that remained firm and unchangeable for many centuries.' That of a celestial hierarchy descending from God, giving Kings (and the Pope) a divine right to rule over their kingdoms and the basis of the feudal system. Cassirer argues that it was only with new thinking in the Renaissance, and most especially with the revolutionary celestial model proposed by Copernicus that this began to change. Cassirer believed that Machiavelli was the first to break away from the entire scholastic tradition, repudiate the divine right of kings and reject the medieval celestial hierarchy. Machiavelli didn't just start afresh though. Instead he used the work of his predecessors, Civic humanist writers such as Leonardo Bruni, Poggio Bracciolini and Niccolo de' Niccoli and they themselves were influenced by the work of writers who are earlier still.
In extremely simplistic terms, the Renaissance can be understood as a rediscovery and revaluation of the art, history and philosophy of the classical period. The writers and thinkers of the Renaissance beginning in Florence were responsible for finding and translating an astonishing amount of the work of Classical thinkers, both Greek and Roman. Machiavelli's arguably most influential work is titled Discourses on the first ten books of Titus Livy, and is essentially a how to of how best to found a republic. Based on the history Ab Urbe Condita Libri (lit. 'Book since the founding') that the Roman republican historian Titus Livius wrote in about 27-25BC. Livy was pretty much the final historian of the Republic, and his history covers the period from the legendary founding of Rome in or about the eighth century BC right up until the very end of the republic, in the reign of Augustus. Livy of course was a product of an intellectual tradition that stretches back across at least the preceding 8 centuries of historical and political thought, from which only fragmentary writings remain (indeed, we only have a fraction of Livy's work).
Further reading, have a look at Pocock, J. G. A., The Machiavellian Moment: Florentine Political Thought and the Atlantic Republican Tradition Quentin Skinner, The Foundations of Political Thought: Volume I & II
As for whether the United States is the world's oldest democracy its not, but its interesting to study why.
Suffrage (the right to vote, and ability to do so) was not universal (for the male population) until the 15th amendment in 1870, the right to vote, and the actual chance to vote is a different thing, (especially with regards to Civil Rights in the southern states) because it wasn't until 1965 when Congress passed the Voting Rights Act that you could call Suffrage universal. France by contrast, instituted universal male suffrage in 1792 with the establishment of the first Republic. Britain did not institute Universal Male Suffrage until 1918, but successive acts during the 19th century had gradually extended the right to vote across a wider and wider section of the populace.
The rights of women to vote is also a consideration and in the US, Female suffrage was not technically universal until 1920, and the same proviso vis a vis the Voting Rights Act holds true.
In Britain Women were accorded the same right to vote as men in 1928, but France did not have Universal female suffrage until 1944.
New Zealand was actually the first democratic country that still currently exists to establish universal suffrage for men and women, when the Electoral Bill was given Royal Assent by the Governor in 1893.
If instead, we ignore the extent of suffrage, and look at countries who have instituted and maintained democratic systems the United States is still not the oldest democracy. The Republic of San Marino claims to have been founded in 301 AD, and became a commune somewhere around the tenth century, its current legal statutes were first codified in 1600, in six books that fulfil much the same function as the Constitution of the United States (albeit as books). This article from the American Historical Review lays out much of the history of San Marino, the world's oldest democracy.