Are they saying "stay with me, cause you're all I need" and "won't back down, no I won't back down" is the similarity? If so, that is ridiculous. I could make a list a mile long of songs with a few similar notes.
It wasn't so much the notes as it is the rhythm and stresses in each lyric. They are incredibly similar but I could see how the writers may have unconsciously put that song together like that. Heck maybe they heard 'Won't back down' the day before the writing session and were channeling it that way. Very interesting nonetheless.
The writers for "Stay with me" say in the article they had never heard of TP song, but the similarities are very obvious, which I think is why they decided to settle amicably.
I don't know if you could get that close by sheer chance though. I guess it's possible but "Won't Back Down" is played so often (in the states) I can imagine even hearing it in a grocery store and pulling it as inspiration.