Do body styles from Pokemon have any effect on gameplay?

Sites like Bulbapedia and Psypokes list Pokemon by body styles, but nowhere is explained when they were introduced or what their function is.

Are the body styles only a way to categorise species in the Pokedex (for example, you can organise Pokemon based on "shape" in Pokemon X/Y) or is there any more to them?

Actually, while Psypokes might not explain it (anywhere I can see with a quick search), Bulbapedia does. If you click the words "Body Style" above the body style icon of any given Pokemon entry, it is a link that takes you to the "List of Pokemon by body style" page which states:

This is a list of Pokémon by their body style according to the Pokédex. This listing was added to the search methods in the Generation IV Pokédex. The game only uses symbols to define each group and without any fixed order so the titles below are simply fan-created descriptions.

Pokémon with multiple forms may be listed differently depending on the sprite used in the main Pokédex entry, which can be changed in Generation V and VI. Currently, the only Pokémon who change drastically enough for this to apply to are Giratina and the Forces of Nature.

So, no, body styles have no effect on game-play. They are just a different way of searching through the Pokedex. Useful if you are trying to assemble a themed team with certain visual qualities, or if you can't remember the name of a pokemon, but know vaguely what it looks like.