I think it depends on the ethnic group and the city. Italian Americans, even those whose PARENTS were American-born (i.e. the grandparents or great-grandparents were the immigrant generation) are likely to identify as Italian/Italian American. More recent immigrants might identify with their direct heritage as well. I dunno.
One thing, which again I'll admit is just based on anecdote or impression only, I feel seems to be is that more Americans (in the Midwest and Northeast I've seen) than Canadians feel or put an emphasis on a genealogical/ancestry-based pride based on heritage, rather than direct cultural/linguistic one.
I've heard more Italian-Americans putting emphasis on descent/roots while more Italian-Canadians seems to emphasize the very
direct cultural aspect (been to Italy and has family there, speaks Italian etc.). I think this may be in part due to Canadian immigrants often being newer but I think the "multiculturalism" aspect of Canada (even if it's ) often puts lived/expressed culture rather than ancestry at the forefront of identity.
It's also almost a cliche that you hear about the American who's 1/16 Irish by descent who's very proud of it, or an Italian-descended New Jersey resident who's never left the country but puts up an Italian flag, whereas I see less of that in Canada, where people who display symbols of ethnic pride really do have closer/more direct ties, and if they don't have those direct ties, they won't claim to identify as strongly.
I also feel like genealogy seems a bigger deal for some Americans than Canadians (eg. It seems like Americans make a bigger deal about being descended from Southerners or Yankees etc. or having ancestors that fought in the Civil War etc. much more than Canadians who care about if they are descended from Loyalists or later British immigrant etc.).
Another example of culture/language mattering more than ancestry I feel is how often an Ontarian who has a French last name but speaks no French at all isn't going to call him/herself "French-Canadian" while an Irish-descended Quebecois or even a Haitian-Quebecois would be seen as more "French" than him/her, while on the other hand I've heard New Englanders stateside claim to have "French Canadian" roots based on ancestry despite having no cultural ties anymore to speaking/practicing French culture (so in the US identity perhaps seems less tied to language). Perhaps English/French bilingualism plus multiculturalism makes language/culture more salient than
ancestry alone. Also it seems like the famous claim that certain Americans like to say about being descended from a Cherokee princess or claiming to be proud of a distant Native American ancestor based on "family folklore", that turns out not to be the case and all upon examination --that isn't as much of a thing in Canada (few people will claim to say so without any evidence unless they really are Métis or have some actual/direct ties to Aboriginal communities).