Can I transfer my AppleCare plan to a new iPhone?

No.

Applecare plus is on the device, not the user.

You can request a refund for the unused time, cancel your monthly billing.

You will have to repurchase Applecare plus to cover the new device.

Cancel or change the payment method for your AppleCare plan - Apple Support

No. AppleCare+ isn't transferable between units, except when the replacement came from a service claim, in which case it's done automatically.

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Hello!

According to this, that will happen automatically.

Have a great day!