The only game I can think of to do this as a part of the main story very well, rather than as an optional “Score relationship points to earn this person as a meaningless prize” is in Trails in the Sky 1+2. Old game by now, but it really stuck with me.

The challenge is to involve character relationships in the story itself, in ways you don’t expect; have that person who fell in love with the main character lie to protect them from a murder charge, and then be arrested for treason; have a villain that takes one love interest hostage, only to realize someone else cared even more about them and is far more dangerous, etc.