The export/import feature in #Fedilab was built long ago. It was meant as a temporary file, to be deleted once the data has been imported.
It's handy because you get all your data back (settings) and you're already logged in.
But the main issue is that both devices share the same token, which breaks push notifications (only one device will receive them).
The quick fix: just log in on the second device to create a new token and register for push notifications again. I will improve this.