My DS loves books and has since he was quite young - but at 2yr3mo I am not sure he is really into "stories." Books yes, stories not so much. At 12 months we got him Goodnight Moon and The Hungry Caterpillar and we looked at pictures, touched objects, stuck fingers in the caterpillar holes, named foods, etc, but didn't "read the story." I think after reading The Hungry Caterpillar for 15 months he still doesn't know what the actual story is about but can say all of the foods in it and count the pieces of fruit. With Goodnight Moon, for the longest time DS really only liked to look through the pictures until he got to the point where the "old lady" is not sitting in the chair so he could say "Oh!" and express his disgust that the mommy was gone (

) and to point at the moon and then out our window at the moon. That was all that "reading" was.
We have some books that are real life pictures (ie/ one that is just called DOG and is pictures of loads of dogs which I got him at 15 months and he LOVED from the first moment he saw it), pop-up and lift the flap books, some more Eric Carle books, etc and he loves all of his books....but there is no focus on the story itself. We mostly get loads of library books and because our library doesn't do any in English we really rely on making up stories ourself (which are different every time) or really just talking about what we see on the pages, etc. but never "reading" them, really and DS does not know the difference.