I’m in Rome, and confused why I’m seeing the Fleur de Lis in St. Peter’s Basilica and other places. What’s the connection with Bourbons and the Vatican?

Nothing. The fleur-de-lis is a commonly used decorative symbol, it was used as a symbol of chastity and virtue before any families adopted it for themselves. It is mostly used as a symbol in connection with the Blessed Virgin Mary and St Joseph

The Fleur de Lis is a common symbol throughout history (I think the first images of one was found at mesopotamia). In modern history it's common in France but also as the symbol of the city Florence. And now I'm not quiet sure, but my guess is the Medici, the ruling family of Florence had a big influence about the Fleur de lis spread around Italy.

Quite a few Popes had the Fleur-de-Lis in their Coat of Arms, maybe you saw those?

For example:

• XIII^th century: Urban IV, Innocent V, Nicholas IV
• XVI^th century: Paul III
• XVII^th century: Innocent X
• XIX^th century: Leo XIII

See here: https://en.wikipedia.org/wiki/Papal_coats_of_arms

Wow! TIL the Fleur de Lis is so much more than the Bourbons! Thank you all for the fantastic answers.

And look at that! Like y’all said it’s on the Medici coat of arms ( they were in Florence) Medici coat of arms

Uh oh! Here it is on the Farnese Coat of Arms as well! Although I thought their thing was bees

Like others have said, the lily you saw are likely related with papal coats of arms. In general, popes often embedded their coats of arms into the monuments and villas they built.

Symbologically speaking, the fleur-de-lys is generally associated with various religious symbols such as the Blessed Mother, St. Joseph and purity in general, which is why we find them so often in heraldry (like you said the Bourbon coat of arms but even before then it's always been a symbol of France since the Frankish king Clovis' conversion), the Italian city of Florence, the French city of Lille, the British House of Lancaster and an uncountable number of Italian comuni.

Interestingly but obviously linked to the Bourbons, you'll find fleur de lys everywhere in Louisiana, even the New Orleans football team has it as a symbol