I have never seen a definitive answer regarding the origin of the name "pizza" but there are as many theories as there are pepperoni on a family-sized ... uh ... pie. Some of the simplest are that it's a corruption/adaptation of the Greek "pita," or that it comes from the Italian musical term, "pizzicato," meaning "plucked," supposedly because when the dish is done it is 'plucked' from the oven.
As to the "pie" part ...
Why Pizza is Called Pie ? The Story of American Tomato Pie
" ... Americans refer to pizza as ?pie? because early 20th century Italian immigrants to the United States made and sold a pizza called a ?pomidore pizza?. This unknown Italian food was translated to ?tomato pie? in English due to being round and cut into slices. As one of the first kinds of pizza sold in the US, the name ?pie? stuck and began to be used to refer to all types of pizza. Today, many New Yorkers still refer to pizza as ?pie?...."
I can't say I'm familiar with all European culinary traditions but all those I have come across that call a dish a "pie," it's a decidedly hearty food, like (Russian) pirogi, (English) shepherd's pie and Cornish pasty, or (Turkish) borek. Pizza, OTOH, was created as a street food for the cash-poor, a lot of bread with a little flavoring on it.
Tarts/tortes, OTOH, are more widespread in Europe and in Italian they're called a "torta" or "crostata," not a "pie." And if there is no concept of a "pie" in the Italian lexicon, that would explain why the word doesn't appear in any dictionaries.
As to the song, "That's Amore," "Dino Crocetti" was born and grew up in Steubenville, Ohio, and the song's lyricist, Jack Brooks, was Liverpudlian, so I wouldn't hang my hat on that as evidence of its authenticity.
RE: the Dean Martin song, he also uses the phrase, "pasta fazool," which the songwriter used as a deliberate corruption of the Neapolitan word for beans, "fasuli," to make it rhyme with "drool." In the most of the rest of Italy it's called "pasta fagioli" (fah-jo-lee) and Dino's family were from Abruzzo, on the opposite coast from Naples. My guess is he used "like a big pizza pie" because that was the best he could do to rhyme with "when the moon hits your eye...."