My Nintendo is a feature of the Nintendo Account system that allows for you to collect rewards for your gameplay. In the game's menu, you can tie your Nintendo Account to your game. This provides a variety of features including bringing in your account's Mii into the game to use as your profile and when you appear in the stages. It also allows for you to be able to transfer your data between devices.
However, like other Nintendo mobile titles, you can collect special Pokémon Rumble Rush Platinum coins which can then be used to get rewards. Some of these missions can be repeated weekly.