This mostly depends on the app developer enabling such capability. There are 2 ways it can be done, both involve using cloud-based saves.
One - Enable Google Drive backups. This works on Marshmallow and Nougat, the app backs up to Google Drive and should restore on the new device, provided both devices are running versions of Android that support this.
Two - Google Play Games. If the app supports Google Play Games, sign in as the same account on both devices and the progress will sync.