Is there any advantage to someone running Ninja Dog difficulty? After a quick look online: It seems like damage from enemies is lighter, items can be cheaper, and Ayane can give damage buffs from her shuriken?

The thing with Ninja Dog is that you need to start that difficulty from loading a save file. It gives you weaker and dumber enemies, free ribbons that have the same function as armlets. But it's a joke difficulty which is why you have to die three times then you can make the save. Plus the game also changes the title screen to Ayane as an added insult. Showing that Ayane is the master ninja, not Ryu.

I will have to double check, but I thought that it input a new difficulty at the start screen?

Nope I am afraid that Ninja Dog is load save file exclusive feature. It doesn't function similar to DMC3.