You should be able to simply sign in with the same account and ensure iCloud settings at turned on for syncing content.
Read this support article to learn how to sign in with your Apple ID on different devices--> Sign in with your Apple ID - Apple Support
For an Apple Mobile device:
1. Open the Settings app.
2. Tap Sign in to your [device].
3. Enter your Apple ID and password.
4. If prompted, enter the six-digit verification code sent to your trusted device or phone number and complete sign in.
If another Apple ID is signed in you may need to enter the password for that ID to sign out of that ID first.
Read this about iCloud
Sign in to iCloud on all your devices - Apple Support