Leonardo was born in Anchiano, a tiny hamlet near the slightly-larger hamlet of Vinci. Ser Piero's family, however, were big fish in the little Vinci pond, and so tagged "da Vinci" ("of" or "from Vinci") after their names.
When he became an apprentice, in order to distinguish himself from other various Tuscan Leonardos in 15th-century Florence, and because he had his father's blessing to do so, Leonardo was known as "Leonardo da Vinci."
When he traveled beyond the Republic of Florence to Milan, he often referred to himself as "Leonardo the Florentine."
Eventually, Leonardo became very famous. He became so famous, in fact, that for the past 500 years he has had no need of a last name (as with "Cher" or "Madonna"), let alone any indication of his father's home town.
In art historic circles he is simply, as he started out in this world, Leonardo.
"Da Vinci" as in "The Da Vinci Code" is kind of cringeworthy for anyone who traffics in Leonardo.