Is it possible to transfer AppleCare from one phone to another?

Yes.
Subject to the restrictions set forth below, you may make a one-time permanent transfer of all of your rights under the Plan to another party, provided that: (a) the transfer must include the original Proof of Purchase, the Plan's Certificate and all of the Plan's packaging material, including printed materials and these Terms and Conditions; (b) you notify Apple of the transfer by sending, faxing or e-mailing notice of transfer to Apple Inc., ATTN: Agreement Administration, MS: 217AC, 2511 Laguna Blvd, Elk Grove, CA 95758, U.S., fax number 916-399-7337 or agmts_transfer@apple.com, respectively; and (c) the party receiving the Plan reads and agrees to accept the Terms and Conditions of the Plan. When notifying Apple of the transfer of the Plan, you must provide the Plan Agreement Number, the serial numbers of the Covered Equipment being transferred and the name, address, telephone number and email address of the new owner.

No. The AppleCare Protection Plan is linked to a specific device. Wjosten's reply and the text copied from the plan documentation (actually, the US plan, which may differ from the one that applies in Spain) refers to transferring the plan along with the device (i.e. if you sell the iPhone, you can transfer the AppleCAre to the buyer). Whether you transfer AppleCare to the buyer or not, you cannot apply it to a different iPhone.

As provided, AppleCare goes with the device not the person.

Ah, thought he said another person, not device...

"transfer the AppleCare from one serial number to another"

I'm reading to quick...

No.
You actually get to find out everything about them since you're the one that has to transfer applecare.

No. If you search by serial number, all you get is the coverage duration. The only way to pull up any personal data on an AppleCare plan is if you search for plans linked to your AppleID (which the buyer will not have).

Along the same lines ... I purchased my phone in July of 2008 along with the Applecare plan so I should have coverage through July of 2010. Apple did replace my original phone once (due to my gps chip dying on me or something) and now I have a new phone with a different serial number. From day one my replacement phone has had three dark spots on the display, but I was just going to put up with it until the end of my warranty period. Since then one of the dark spots has developed an occasional stuck pixel. Checking my warranty status on Apple's site shows that my phone is no longer covered under warranty. I am correct when I say that my new hardware should be covered under the remainder of my extended warranty - right? I'm assuming the guy at the genius bar just did not transfer it over.