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?
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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.