I think we can safely rule out the idea that Machiavelli was just trying to flatter the Borgias, because when he wrote The Prince, Roderigo Borgia had been dead for about ten years and replaced by an enemy, Giuliano della Rovere, the Pope Julius II. Julius II himself had died and been replaced by Giovanni de Medici (Leo X). If there is any open flattery in The Prince it is in the discussion of Ecclesiastical Principalities, which concludes by saying in regards to Leo X that ["as those popes [Alexander and Julius] made it great by force of armies, so he through his goodness and infinite other virtues will make it both great and venerated."](https://en.wikisource.org/wiki/The_Prince_(Ricci)/Chapter_11)

The reason for this uncharacteristic line is perhaps found in another bit of context--the book itself is dedicated to Lorenzo de' Medici (not that Lorenzo) who in 1513 was obvious successor to Giuliano de Medici, then ruler of Florence, although to an extent they were proxies for Giovanni who could not officially hold power in the city. In short he was writing in a very Medici dominated environment, which he himself had suffered under, being first driven from his position within the Republic, then imprisoned and tortured under suspicion that he was pat of an assassination conspiracy. The Medici were not exactly staunch opponents of the Borgias but Giovanni had spent the time of Alexander's rule in a sort of political exile, and was elevated by Julius II, so it is fair to say the families were not allies. It also may be worth noting that Lorenzo (to whom The Prince was dedicated) was the Duke of Urbino, and Urbino had famously suffered under Cesare but I have not personally seen anyone make anything of that.

Apologies if that is a lot of names there, Renaissance Italian politics could be quite complicated, but the important takeaway is that the Borgias were out, Roderigo and Cesare were both dead, the Medici were in, and the Medici were somewhat opposed to the Borgias.

Tying all this together--Machiavelli's biography and republican sympathies, the discordant sentence about Giovanni, the political situation unfavorable to the memory of the Borgias--and The Prince has at times been interpreted as a political satire, perhaps most famously by Rousseau, who argued that the choice of Cesare as the "ideal prince" was meant as an attack on princes as a whole. But he is not the last and you will often seen people asserting The Prince is satire as a sort of obvious fact, an Early Modern Starship Troopers that you have to be a real dumb dumb not to get. The problem is that the interpretation of The Prince does not really work with any real scrutiny. Often people will juxtapose it to Machiavelli's magnum opus, the Discourses on Livy to say that see, Machiavelli loved republics and hated monarchy. This may be true, but if you read both books they clearly show the same worldview, are very similar in methodology and style of interpretation, and most importantly cross over a bit. Most notably, in Discourses I.9, Machiavelli has a section on how it is necessary for founders of republics who need to make a break with past institutions need to rule alone, and this resonates greatly with the themes in The Prince. There are other resonances as well, particularly Machiavelli’s populist instinct (in chapter 9 he says a prince should put his faith in the people, not the nobility) and his famous hatred of mercenaries. There is also just too much solid advice in The Prince to really be a satire–if he was trying to condemn princes through an obviously evil vision of the ideal, why would he say it is necessary for a prince to avoid taking his subject’s women and property? What is the satirical value in saying that one should not take flatterers seriously?

So if the book is not a satire, and the Borgias are long out of power, why use Cesare Borgia as an example? The best answer is probably a somewhat boring one: while serving as a republican official Machiavelli had been sent as an ambassador to Cesare, met him personally, and found him to be a useful example for his work.