Dragon Quest V, I believe, does something like this. This is a fairly major part of DQ5 that will decide who one of the more important party members in the second half of the game is. SO2 also had a rather complicated relationship meter that was invisible with the main character and their party members.Phantasy Star III has an incredibly primitive generation feature where you make exactly one choice per generation about who to marry that branches the story. It's debatable if the 'romances' in most Baldur's Gate-type games count, since they're usually superficial/LARPing. It was fun to figure out, though -- you could literally see the difference during battles.Now at a public school, Kaede decides to join a club t0 avert the boredom of the classrooms.Since the idea of a swim club keeps popping into his mind, he seeks for it and ends up meeting two girls by the pool: the relaxed and observant Hirome and the wary and short-tempered Mieko.It does involve stat and attributes training/allocation in order to get specific tasks done however and features a very well-written story IMO for an independent developer.It's not really the main focus, and the dating aspects totally vanishes about 1/3 - 1/2 of the way through the game, but FFVII allows you to date Aeris, Tifa, Yuffie, or Barret depending on your actions.
I had one run through the game where none of the girls liked Chisato at all -- they refused to use any healing spells or items on her whatsoever (which was messed up but really funny).This list headlines the best dating games ever made.