There’s a couple ways to answer that. The first being looking at the US only. And even these can be subjective based on how detailed you want to get, or how much of an area you want to talk about.
There’s the New England states - most of the north eastern part of the US
Mid-Atlantic - Between New England and the South
The South - Roughly North Carolina south, and then west to Louisiana or Texas (depending on how one views things)
Florida is just Florida
Texas many times is just Texas
Ohio Valley - The states that surround Ohio from the New England states, to the Midwest, and generally abuts the southern states
Southwest - Texas, Arizona, New Mexico…possibly Colorado and Utah as well
Midwest - the states between the Ohio Valley/Southern states to the Rockies
Mountain states - The states predominantly along the Rockies
Pacific Northwest (PNW) - Washington State and Oregon generally, sometimes Idaho as well
California - usually divided into Northern Cali and Southern Cali
These can be highly subjective, some of them can be condensed into other sections if you want to speak in more broad categories - the ‘major’ areas being the Northeast, the South, the Midwest, and West. There are even more specific regional names like the Appalachians, Northern Plains, Great Plains, Great Lakes, Old South, Pacific, etc.
Texas many times is, depending on context, is given its own group by itself as Texas is a rather unique state, as large as it is, it was its own country for a time after breaking off from Mexico, and incorporates many factors that can associate it with surrounding regional titles given its size and location. Florida is often times not considered part of the South (with the exception possibly of the panhandle) given that it was owned by the Spanish until 1819, and did not become a state until 1845. Given the strong history and current influence still of both Spanish and Hispanic cultures in that state, and that they don’t express the same traditions typically, it is usually considered its own ‘area’ so to speak.
No, Hawai’i is not part of the continental US, you are very much correct on that. So when referring to the Americas, Hawai’i would not be considered part of that, as its a Pacific Island. Though when referring to ‘American’, it would be considered part of that term as it is still part of the US.
The second way to answer that is many times, while someone from the US will say they are American, in some ways, you could draw parallels in saying that each state in the US its own country in being analogous to European countries. You are European as a larger identifier, but you are also French, Spanish, British, Czech, Greek…whichever country is the one you consider home or are from. I would say I’m American, but in similar terms could identify as saying I’m from North Carolina, Alaska, Nevada, Maine, etc. The smaller part of the larger whole.