Actually, this kind of "double が" situation happens all the time. Sometimes there's just no elegant way around it. In this case, Aさんは doesn't feel exactly right because these notifications appear out of the blue with no surrounding context. In cases where brand new information is coming in, が usually feels better in introducing it. Imagine it sort of like an announcement of a radar contact: "Sir! A meteor has been detected on the starboard side!" This would definitely be が. The Facebook notification is very similar in feeling.
I admit that I too feel a tinge of worry when I'm forced to use this sort of construction, but I've never seen a native speaker have any trouble understanding these.
Edit: Sorry, I didn't answer your actual question:
Why couldn't it be read as both A and B liking the photo?
If both people liked the photo, the sentence would be more explicit about that. It would look something more like:
On the other hand, if only B liked the photo that B himself shared, it would like be simply:
The fact that two parties are mentioned makes the reader expect that one of them lives in a relative clause. So it is natural to expect that A is the main subject of the sentence.