First of all, "Napoleon" is a very unique name, so it's not too surprising that it's stuck. If his name was "Jean," I imagine matters might be different.

More importantly, though, there is one important group of people in post-medieval history who do tend to be known by their first name: monarchs, who typically have a regnal name. (Popes are a similar situation.) So the man who contested the rule of France with Napoleon was known as "Louis XVIII" or just "Louis" by his contemporaries, not "Louis de Bourbon" some other combination of his personal names and family name. Louis was an incredibly popular name in France at the time, and yet still people casually referred to "Louis" when the context was clear that they were talking about the reigning king. (If it wasn't, people specified.)

Now, this is relevant for Napoleon because on Dec. 2, 1804, First Consul Napoleon Bonaparte, legally a common citizen holding a high government office, was crowned as Emperor Napoleon I of the French. That is to say, his official regnal name became "Napoleon."

Until this time, people called him both "Napoleon" and "Bonaparte" (or "Buonaparte," especially when they wanted to be snide about his alleged lack of Frenchness), but speaking casually Bonaparte seems to have been more common, and a search in Google's Ngrams backs this up: in 1800 the word "Bonaparte" (case-insensitive) was used about five times more often than "Napoleon" in the English-language corpus; so was "Buonaparte." After about 1810 or so the word "Napoleon" starts to rise in popularity, becoming the most popular of the three Bonapartist terms in 1811 and staying that way ever since. The French corpus tells a similar story (especially if you remember to search for "napoléon" and not just "napoleon," apropos of nothing) except "Napoléon" becomes the dominant term in 1805 instead of 1811.

The other factor we need to consider is that there were lots of prominent Bonapartes tagging along in Napoleon's wake. They became especially prominent after he crowned himself emperor and started making his brothers and sisters kings and queens of various European countries. So right as Napoleon is pushing "Napoleon" as his official name rather than "General Bonaparte" or "First Consul Bonaparte," a bunch of other lesser Bonapartes appear on the world stage. To be sure in most contexts in the period 1796-1821, any reference to "Bonaparte" would have been instantly understood to refer to The Man and not Joseph or Louis or Lucien Bonaparte. But they were all Bonapartes; there was only one Napoleon.

Finally, speaking from the 21st Century and not merely the early 1800s, one must consider Louis-Napoleon Bonaparte, the greater general's nephew who crowned himself Emperor Napoleon III of France in 1852. ("Napoleon II" was Napoleon's only legitimate son, born in 1811 and recognized by Bonapartists as the rightful emperor of the French from Napoleon I's death in 1821 until his own early death from tuberculosis in 1832.) Like his uncle, Napoleon III used "Napoleon" as his regnal name (even though his given name was "Louis-Napoleon" — a common practice; Louis XVIII's given name was Louis-Stanislas-Xavier). Napoleon III consciously drew on the mythos and popularity of his more famous (and more able, though Napoleon III was certainly not talentless) uncle to bolster his own popularity. In any case, you had another head of the French state using the resources of the state to promote "Napoleon" as the key name, rather than Bonaparte, reinforcing the trend that we can trace back all the way to the early career of General Bonaparte.

I don't want to discourage any other answers, but this question was answered in a few threads on this sub:

An answer by /u/DonaldFDraper: Why is Napoleon referred to by his first name?

An answer by /u/kieslowskifan: Why is Napoleon commonly referred to by his first name, while most other leaders are called by their last.

An answer by /u/JehovahsHitlist: Why is Napoleon known as "Napoleon" and not "Bonaparte"?

Follow-up question: Why was Michelangelo Caravaggio known as "Caravaggio" and Michelangelo Buonarrotti as "Michelangelo"?