Replace the words
'faceless man' with the words
'top assassin'.
He was training her to become a
'top assassin' first & foremost.

Not the other way around.
So if you are a 'Top Assassin'... then being a 'faceless man' is just the BEST WAY to completely
'conceal your identity' before & after you go on a mission to kill a 'target'.
You don't necessarily need to be an 'assassin' to be a 'faceless man'.

But it HELPS ALOT to be a 'faceless man'... If you are an 'assassin'.
If Arya could kill another (fully-trained) assassin who is using the 'best techniques' in the whole land to eliminate her too at the same time.... someone who is also bigger & stronger and a MUCH better 'combatant' than she is... then there's really nothing left that Jaqen could teach her about becoming a 'top assassin'.
- she learned how to use additional weapons
- she learned how to stalk her targets
- she learned how to identify their weaknesses
- she learned how to live & fight well in total darkness
- she learned how to evade her enemies
- she learned how to 'hide in plain sight' (when she was a blind beggar)
- she learned how to use other people's faces & assume their identity
- she learned how to identify poisons 'by smell'alone
- she learned how to slip said poisons into her target's food without getting caught
- she learned how to lie convincingly
- she learned how to control her emotions
- she learned how to 'spring a trap' on a target
- she learned how to use the 'environment' to her advantage
- she learned all the rules of the many-faced God
Shhheeeiiittttt... That should be enough to get a
'diploma' from Shiesty State University of Braavos.
The only 'credits' she prolly missed was for the Assassin-level class on making
'smoke bombs'.
Just saying.
The House of Black & White.
You either 'graduate'... or
'die trying'. 
Literally.