Yeah I re-read it now and don't feel so strongly about it. But I think its the sentence structure:

"This is the longest a pig organ has survived in a living person"

The pig organ is the first thing mentioned, with the living person last.

Whereas:

"This is the longest time a patient has survived after a pig organ transplant."

This puts the patient first, and doesn't give as much of that kind of vibe.

Edit: Actually, that updated sentence may need some adjustment. I assume when organ transplants fail we don't just let people die with them. So maybe that's why the original sentence had such emphasis on the survival of the pig organ...