"Genius" in the classical sense means "spirit". It's distantly related to the word genesis, meaning "birth", as well as to words like genus "kind", genre, and gender. Another related term you might here is genius loci, a protective spirit of a place.
It's a little hard to wrap all of those terms into one concept, but you can see where it's going. They're all in the vein of referring to the thing that makes something individual: its birth, its kind, its spirit.
That's the sense in which Blake means the term: the true spirit of a human being, derived from the sense of poetry. Poetry here is used as a stand-in for knowledge in its pure essence, the "true" meaning of words, setting aside the confusion that human beings layer on the in the same way that the difference of bodies obscure the spirit in which all human beings are one.
The sense is that poetry gets its power from being closer to true meaning than prose. That's a little hard to feel today, perhaps, but you've certainly felt the way a song can move you. Poetry had that same feeling for Blake and other romantics: there's a "rightness" to it which moves you in ways that go beyond the simple surface meanings of the words and grammar.
So the "Poetic Genius" is that spirit which binds all humans, the Platonic truth, of which every philosophy and religion is a noble but flawed shadow.