Why is it called an egg cream? I noticed there are no eggs in the recipe. I would have thought of this as an Italian soda. What is the difference?

There's a good explanation here as well: https://www.merriam-webster.com/wordplay/egg-cream-history-name-origin

Yes to what HalfPint wrote.
And here are some (apocryphal?) origin stories from Wikipedia.
https://en.wikipedia.org/wiki/Egg_cream

Traditional egg creams contain neither eggs nor cream. An egg cream usually has milk and soda water (seltzer), mixed with a flavored chocolate or vanilla syrup. One story of its origin is: It was a soda fountain drink that did originally have eggs and cream. In poorer neighborhoods, a less expensive version evolved that only had soda and milk. An Italian soda is syrup and soda water and no milk (as far I know).