【Updated: 】
It is possible to continue using Fire Emblem Heroes as the same user on another device if you link the application to your Nintendo Account.
After linking the application on the original device to a Nintendo Account, launch the application on the new device and select Create/Link a Nintendo Account.
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?"