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As for the topic at hand, the non-Earthrealm "ninjas" look like ninjas, because [and this is just my take]: 1.) They look 'cool' (or so many feel) 2.) They're assassins 3.) It doesn't stray too far from their original designs (aka palette swap ninjas)

Don't focus too much on the veracity of a character's clothes, as it's merely stylistic choice.

Ermac, Reptile, Rain and Chameleon are all assassins/warriors of Shao Kahn. Therefore, they wear the ninja-assassin garb, much like the Earthrealm "ninjas" do.
However, note there is only one ninja in MK: Scorpion.

And as for if any of the Outworld ninjas ever trained under the Shirai Ryu or Lin Kuei, I'd say the answer is 100% no.

Ermac was 'born' in the midst/slightly before the MK1 tournament, so that's definitely impossible for him. Also, Outworlders/Edenians already have mystic powers, are stronger (probably) than most Earthrealmers, and probably have plenty of other skilled native warriors to train with.

They don't need Earthrealm ninja clans, that is, unless they're really dying for a new pair of clothes.