I think it does have to be done in one run, but my brother's the one who did the first playthrough, and he rushed through the game on the easiest difficulty setting using only the original four characters, so that's no help. I think I was at around 250 lines, though, so I don't think the game was counting any of the unique boss lines from later in the game.
Some people claim to have earned the trophy while just playing normally and without having to keep track of Rise's lines, but that seems crazy to me. I kept an alphabetical list that I added to every time she said a new line. I found it very useful for things like status effects, where I wasn't sure if I had the lines or not. I felt like I hit the jackpot when I came across and enemy with Debilitate and realized that I could get 21 unique lines just from that.