I've known a few people who had names that were difficult for most Americans to pronounce or spell, and so used "American names". Or maybe they were just concerned about having a name that "sounds strange".
There's nothing illegal about using an invented name as long as you are not attempting fraud. (I'm not a lawyer, maybe there are some other specific cases.) Plenty of native-born Americans use nicknames.
Whether this really helps in getting jobs or whatever, I don't know.
If you have email addresses and the like in your real name, you could get new accounts with your "American name". Otherwise it's likely to be confusing. In the case of an email address, you may be able to set it up to forward to the email with you real name. For most sort of user names that's not an option of course.
If you have job history under your real name that you want to use for references, yes, you'd have to tell the HR person or whomever your real name.
All around, it seems like a pain to me. There's no easy solution to the problem of avoiding confusion if you're going by two different names. Enough people do it that I doubt anyone would question it. Like, few would say, "Why are you using a fake name? Are you trying to hide something?" But you'll have to tell them your real name sooner or later, so if you're thinking that you're going to hide the fact that you're from Mongolia or wherever to avoid being discriminated against by anti-Mongolian racists, I doubt that would work.