There's nothing in the episode or the context to liken the apparent callsigns to any registry scheme we have ever heard of.
Quite. I'm guessing that Starfleet knew all along it was eventually going to have this brand new NX class. That was what they were building up to, so to speak. So the test vehicles intended to prove that its engines could work, were called NX-Alpha and NX-Beta, because they were always intended to "lead" to the NX class.
It'd be like (a few decades down the line) the UFP Starfleet deciding to produce prototypes called "Constitution-Alpha" and "Constitution-Beta" to test the engines that would one day be used on the class which they were already planning. That is literally the only meaning the 'alpha' and 'beta' designations have here. And there is zero proof to the contrary.
And I do like the 'U.E.S.' designation for Earth Starfleet ships (though I think NCC sounds better as a registry prefix). I always wondered why Earth Starfleet vessels had no prefix - just the name.