Does AppleCare warranty follow the device or the user? I've bought a used MacBook Air and according to the serial I've noticed that the previous owner has linked an AppleCare plan to it, up to 2017.

Does the warranty follow the device or the user? Better put, as the new owner, can I take advantage of the warranty or is there a need of some sort of transfer?
The previous owner has to transfer it to the new owner.  See, "Is the AppleCare Protection Plan or AppleCare+ transferable?" under AppleCare Products Frequently Asked Questions, it states:


  Is the AppleCare Protection Plan or AppleCare+ transferable?
  
  Yes. If you choose to sell or give away your Apple product, you can
  also transfer the ownership of the AppleCare Protection Plan or
  AppleCare+. Please see the Terms and Conditions for complete details.