My reaction was exactly the same as Ed's. It sounds odd to my native ear, too, and I'd also say 'We play games on our phones'. I did find quite a lot of references to 'phone games' when I googled it - so maybe some (young?) people do say this. It's probably safer to stick with the phrase which we all agree is right, though, just to be sure.

Yes, its fine saying that!

I've never heard it said that way, and to my native ear it sounds a bit strange. I'd say: we play games on our phones.