I think that we are overthinking this a bit. Let's try to be rational about it.
Q:What is the way Braavosi recognise Faceless Men?
A: Iron coin and the words "Valar Morghulis".
It stands to reason that there is limited number of iron coins. The only way to get one for yourself is to either kill Faceless Man (or loot him), in which way you either won't know the words (and be killed on sight), or will really want to and be able to join. The other way is to be given the coin and told the words.
I don't see a child killing a Faceless Man or dreaming to become one, especially not Braavosi child (who know the words and respect FM). That leaves option number two: get given a coin and told the words.
Now, Jaquen could have sent a coded message to Kindly man saying something like:
"Yo Housekeeper, I have found a new recruit.
She is fierce, high-born, quick thinking and, all in all, a very promising young lady. She is Arya of House Stark, but goes by the name of Arry or Weasel or, by the time she reaches you, some other names.
Her name doesn't matter, because she currently looks like this: [Arya's short description, how it changed over time, including the little Braavosi styled sword on the hip].
She knows the value of life and almost killed me with a whisper [short explanation of what happened], so I gave her one of my coins (making the number of them currently in my possesion four) and told her the words. Expect her in a year or so, because she is still trying to go home to Winterfell.
Sincerely yours,
The Fat Man
P.S. That thing in KL was finished successfully, moving to do the Iron Islands project next. "
Then he finds the closest Braavosi ship, shows the coin, tells the words and gives them letter to be delivered to The House of Black and White.
The kind man looks at peerage books, or FM files if you will, finds Westeros, finds House Stark of Winterfell, finds Arya in there and makes a note.
A year or so later, a kid enters the house (which you can't do unless showing the coin and telling the words. maybe coin is worth a life or free death (suicide?)), wearing small Braavosi stile sword, helps a nobleman to die by giving him a drink and then kind man shows up and talks with her. One of the first thing he tells her is that she is too young to look for mercy of many faced god, so she must be something else. Then she shows him the coin and the above exchange happens (quoted by Imp).
No need for anything supernatural or magical.