I assume when you say "American culture" you are referring to the "United States of America culture" not one of the other countries in the Americas.

This has to be one of my least favorite bits of pedantry out there.

Everyone is aware that there are other countries in the Americas, the US is the only one that is commonly called America and the people who live there "Americans," and in fact none of the others have the word "America" in their name.

The only time I see people trying to refer to anything besides the US as America/American, is when someone feels compelled to bring up this point. In actual usage, people will refer to the things as being North and/or South American, or "From the Americas," or in certain contexts, they may use "New World." Otherwise, they'll refer to specific countries or regions, like "The United States and Canada," "Central America," "Latin America," "The Amazon Basin," "The Pacific Northwest," etc.

Because frankly, there aren't too many contexts in everyday usage where it's useful to lump both contintents together as a whole, the two continents have a pretty diverse cross section of different cultures, languages, economies, climates, geography, seasons. One of the only things you can really say about them as a whole is they're not physically close to most of the "old world" (Europe, Asia, & Africa) and even that's technically a little iffy because part of Russia is pretty damn close to Alaska.

Because really, what sort of useful comparisons are there to make between, say, Newfoundland, Kansas, and Peru besides to say "Yeah, those places sure are all different from each other?" What do you gain by trying to lump them together?