It was the same in Champions Online and it is the same there; the poster who noted that its for the purpose of allowing you to have any name you want is correct. It also allows folks to -cross-talk- into the other game (CO). This may not be the way some would implement this feature, but its how they did it here and its part of their core-structure, so don't anticipate a change to it anytime soon.
Then what is the point in letting you choose a full name for each of your characters if NOBODY can see it???
That is so stupid.
That's incorrect; anybody who wants to can see it. It's in your info page for the character.
The point of the global handle is to allow friends lists that work the way everybody always complains they don't work in other games; add your friend once and have all his characters. Or equally importantly, add someone to your ignore list, and have all his characters.
It's not just global to STO; it's also your unique identifier in all Cryptic games and websites, including the "super secret" next game they're working on.