I've read that shiny Pokémon now exist in Pokémon Go.
See for example this article.
If I have a regular Pokémon and a shiny Pokémon, both with the same IVs and attacks, are they identical in terms of battling strength?
Both Pokemon will be identical in terms of battling strength.
As in the Pokemon games, the only difference between the a shiny Pokemon and a normal one is the fact that the color of the shiny one is different (adding a glow effect too). But the logic of the stats of the Pokemon remains the same. So a shiny Magikarp wouldn't be stronger than a normal one.
It has been mentioned in this article that:
The shiny Pokemon are not actually inherently more powerful than their less shiny counterparts. In fact, the early evidence suggests that there’s really no link between a Pokemon’s shininess and its individual values.