Is it possible to continue playing Fire Emblem Heroes as the same user if I change devices or obtain a new one?

You can use the same save data on your new device by linking the save data with which you are currently playing and your Nintendo Account ahead of time.

■ See the following for instructions on how to link your save data.
How do I transfer my save data? (When changing to a new device, etc.)

Related questions:
"If I reinstall the Fire Emblem Heroes application, will the Orbs I've purchased or Heroes I've summoned disappear?"
"If I change devices, will the Orbs I've purchased or Heroes I've summoned disappear?"