Obviously not. But I’m joking to bring up a point.
Vatican City is a sovereign country. I think Mussolini (ironically) signed the treaty. But it is also properly part of the city of Rome, which is Italy.
How can it be both?
But it’s not part of the city of Rome. (Except geographically)
Because it was founded in Rome, and based in the area of Rome.
Do French fries and French toast belong to France?
The Vatican is the sovereign seat of the Holy See, the governing body of the Holy Catholic and Apostolic Church, commonly referred to as “Roman” because of historic origins.
It’s not “properly part of the City of Rome” any more than Beverly Hills or Inglewood are “properly” part of Los Angeles, the District of Columbia is part of Maryland, or more to the point, any more than Monaco is properly part of France.
Most people who work at the Vatican commute in from Italy or are on a time-limited work assignment, and remain citizens of their countries of permanent residence. Of those who are resident at the Vatican, about half have a Vatican passport.
So is the entrance and exit to Vatican City like any other border crossing? Does an Italian citizen who works at the Vatican have to show his passport as he enters Vatican City, and get his passport stamped? And then when he goes home show his passport again when he re-enters Italy? Or is there some sort of special arrangement that allows people to come and go without passports?
Like in a lot of the continent, you do no clearing of immigration or customs to cross; they’ve been in that status under their mutual agreements even before the EU/Schengen free-transit zone existed (Vat City is a Schengen “associate”).
My understanding is that we use the foods under a long standing treaty. Escargot as well. One of many reasons I hope we never go to war with France 
Freedom fries for everyone! 
Belgians say “french fries” belong to them.
I thought that sprouts belonged to them.
I’m not seeing the joke here.
Nor is it obvious that the Vatican does not have dual citizenship. On the contrary. Vatican law does not prohibit dual citizenship and virtually everybody, if not actually everybody, who has Vatican citizenship also has the citizenship of another country.
Which is kind of a big exception since, in most contexts, the social-geographical reality that is a city is of rather more significance than the minutiae of local government arrangements and boundaries.
Do you know where the boundaries of the City of London lie? Probably not. Do you care? Almost certainly not. Most of the buildings and institutions that you associate with London lie outside the City of London - Buckingham Palace, the Houses of Parliament, the British Museum, the Tower of London. Does this matter? Not at all.
Only in Brussels, the rest of the country it’s waffles.