The Label is currently the only way to organise characters, but that does mean it ends up rather personal and irrelevant to others.
Let's look at two scenarios:
1. The GM is looking at the characters in the game to make an appropriate mission from their backstories.
2. The player is looking at all their characters at the same time (I don't know why).
In Scenario 1 the GM sees either
"(DnD5e Fighter) Bob | Character Sheet | Gamers Plane"
"(DnD5e Fighter) Barb | Character Sheet | Gamers Plane"
and probably only sees the first few characters, so they all look alike.
Or
"Bob | Character Sheet | Gamers Plane"
"Barb | Character Sheet | Gamers Plane"
In the Second scenario, the players sees either:
"(DnD5e Fighter) Bob | Character Sheet | Gamers Plane"
"(TftL) Frank | Character Sheet | Gamers Plane"
Or
"Bob | Character Sheet | Gamers Plane"
"Frank | Character Sheet | Gamers Plane"
In the first, as things stand, that is a problem for the GM (or any players looking to use the existing characters to help build their own). In the second case, everyone sees something useful, with the exception being when the player both has many characters with the same name, and wants to see them all open at the same time.
I can not see any value in using the Label. I also don't care much about this, it is only a minor inconvenience. If anyone has reasons to want the Labels, I won't stand in their way.