The Enterprise was the NX-01, of the NX class
And if pennant letters in the UE Starfleet mean the same thing as they do in today's navies, that is, they describe the role of the ship, NX might well be the designation of deep space Xplorers - of which the Enterprise
would indeed be the very first ever.
There might have been, say, five different CC classes of combat cruisers, the first featuring CC-01 through CC-12, the last featuring CC-78 through CC-94 or something like that. But only one NX class. Which then means that "She's NX class" would be a workable, unique identifier, an alternative to "Enterprise
class" - but nobody could say "She's CC class" and get his message through, hence the need for "Neptune
class" or "Triton
class" when describing combat cruisers.
From the Trek history we've been presented with, the New United Nations and some version of United Earth did exist at the same time.
I don't think we need to presume this. The only date at which the NUN was known to have existed was 2036, and our earliest UESPA reference would seem to be from the late 2060s; what other UE references are there for the 21st century?
Word of Bob Orci > T'Girl
...But word of Bob Orci << word of T'Girl.
Beyond that, Kirk as played by Pine looks different from Kirk as played by Shatner. The two aren't of the same height, either, nor do they have the same eye color. Basically, this means that according to Bob Orci, the USS Kelvin
we see may be the new look for the original USS Calvin
which was more angular and twenty percent smaller but was still the very same ship.