Ryoko's Guide to the Yokai Realms has a Bender class (or at least a 5e interpretation of the Benders from Avatar: The Last Airbender and The Legend of Korra😛). So, how about a Monk/Bender multiclass build? ;) The Monk's martial arts plus the Bender's ability to manipulate the elements.
Several other posters on this thread have already brought up this particular multiclass build. ;)
Here's a multiclass build for the Ninja that I came across recently:
http://5emulticlassing.wikidot.com/ninja
It also makes use of the Shadow Monk/Assassin Rogue build everyone here has been...
What about that upcoming Monk subclass, the Way of the Mystic? It sounds like if it was multiclassed with the Rogue class, it could also make for an interesting Ninja.
D&D has it where background equates to whatever job/profession you had prior to setting out and becoming an adventurer. Since historical Ninjas were known for their stealth/infiltration/deception, it's more likely they would be members of a background that would let them blend in with the rest...
Ryoko's Guide to the Yokai Realms by Loot Tavern does have a Shinobi subclass for the Wizard. As a wizard, the Shinobi could pick spells that mentally affect their targets (Genjitsu), buff up their natural abilities (Taijitsu) or manipulate an element (Ninjitsu)
Especially since PF1 already has a number of hybrid classes such as the Arcanist (Wizard/Sorcerer), Brawler (Fighter/Monk), Slayer (Ranger/Rogue), etc.
I would go for the Shinobi in the Naruto anime for this. Give them a feature where they get to start out with an affinity toward one element (air, earth, fire, lightning and water) and pick up several more as they level up. With regards to subclasses, there could be one for those players who...