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...

Seeing as the subject of the sentence is the pig organ, it's saying the organ is the one doing the surviving and only tangentionally mentions the person surviving by calling then living. I (and presumably we) only come to a different interperetation because I have the context that the latter is the important bit. If I give some other similarly structured sentences but without context, how would you interperet these?

- This is the fastest Alice had driven since Bob broke the speed limit.

- This is the oldest tree still standing in the burnt forest.

- This is the most stable chemical additive to our long-lasting concrete.

Without context, to me these examples sound primarily about Alice going fast, the tree being old, and the chemical being stable. But if those appeared in articles about traffic law, natural disasters, and sidewalk design, then these phrasings might be less ambiguous if flipped (as another commentor pointed out).