Warranty issues on replacement iphone
It seems the seller has a decent understanding of things and what you wrote smells of no bull. 

Apple warrants goods in the country of purchase (which sets obligations on the warrantee and the warrantor). Do keep in mind that the original purchase date of the phone sets the warranty. A replacement phone would have the remainder of the original one year warranty and a 90 day warranty from the date of the exchange - whichever is longer on it's own (the 90 days isn't stacked / it runs concurrently with any remaining warranty).

Possession means you can register the phone and seek all service as if you were the original purchaser with a few exceptions. You probably can't return it and you would need a notarized bill of sale and original receipt to do things like bypass activation lock. 

You can check your phone warranty status at https://selfsolve.apple.com/agreementWarrantyDynamic.do

You can add AppleCare just as the original buyer could. You can seek any and all other warranty services as if you were the original buyer without needing any proof of ownership or purchase.