Not unless you're European. Bluegrass (despite having a species named after Kentucky), fescue, bermudagrass (which is an invasive species in Bermuda, BTW), etc. all come from Europe.
They're also not very suitable for manicured lawns (as opposed to meadows or praries), which is why we imported the European grasses in the first place.
It depends on the variety of grass. There are some low growing ones that are well suited for lawns. The main argument for the European ones is that they take hold and spread much more readily, because they are invasive.
So there is no grass in natural environments in America?
Not turf grass that can survive being mowed to 2" high, no. All our native North American grasses either grow in clumps (e.g. crabgrass) or need to be more than a foot tall.
Also, the natural environment in the eastern half of the US (where most of the population lives) is pretty much 100% forest.
There are native grasses that can be kept as maintained lawns, such as blue grama (although the recommendation is to cut it a little taller, 3-4"). It's not going to be emerald green like Kentucky blue, however if you live in a dry area with watering restrictions your lawn will be the greenest on the block for sure!
That entire page makes no mention of blue grama being resistant to foot traffic, which is half the point of a lawn and therefore an important requirement for turf grass. The omission (in combination with it being mentioned as slow-growing) makes me suspect the worst.
It mentions that you're supposed to be careful not to over water it, but also that you need to water it more often in order for it to form a uniform turf instead of growing in clumps. Sounds kinda fiddly to me.
No, bamboo would not be native to Europe. That's like saying "There are rats in Europe. Therefore rodents are native to Europe. Therefore Capybaras are native to Europe".
"Poa pratensis commonly known as Kentucky bluegrass (...) [is a] species of grass native to practically all of Europe, North Asia and the mountains of Algeria and Morocco. It is a common and incredibly popular lawn grass in North America (...) despite the fact that it is not native to North America. (...)When found on native grasslands in Canada, for example, it is considered an unwelcome exotic plant, and is indicative of a disturbed and degraded landscape."