Yes. Save data can be shared between the Nintendo Switch version and mobile version of Pokémon UNITE.
In order to share save data between the two versions of Pokémon UNITE, you will be required to log in with a linked Nintendo Account or a linked Pokémon Trainer Club account when using a new device to play with existing Pokémon UNITE save data.
Please note: When you open Pokémon UNITE for the first time on a new device, starting a new game without linking save data will create new Pokémon UNITE save data. If this happens, you will no longer be able to link your existing save data. Always pay close attention and check all relevant information before linking your account.