Am I eligible to claim AppleCare when it was bought by a different user?

i bought an iphone for someone as a gift. i also bought apple care under my name. The person has their own apple ID which they are using for the new phone: does the apple care that i bought have to match the name of the person who received the gift. I bought an iPhone as a gift with my name. I did the and with an apple care. The person using it will use their own apple ID. Am I still eligible to make a claim on behalf of they person as the apple care is in my name.

[Re-Titled by Moderator]