There are a few good ones like that, though not many. Most games, especially RPGs usually only have one main story line and main character. Here's the ones that I've played and can think of off the top of my head. I'll divide them between different characters and factions:
Shogo: Mobile Armor Division
Tactics Ogre series
Seiken Densetsu 3
Sword of Mana
Lord of the Rings (GBA series)
Live A Live
Rudra no Hihou
Nekketsu Tairiku Burning Heroes
Record of Lodoss War (SNES version)
Final Fantasy III/VI
Some games, of course have the option of choosing different characters and sides as well, but I listed them based on which was more important overall. For example, in Der Langrisser, Shogo, Gothic and Risen, the choice of which side you join is the player's choice, but the main character is the same person. Whereas, in Lord of Rings: The Third Age, the choice of character determines which side you're on. Hopefully, this list will prove to be useful to someone.
I didn't bother adding games such as 7th Saga, Ancient Magic, Arcus Odyssey, Crystal Beans, Shining Soul series, Pokémon series and Mystic Ark to the list because even though you can choose which character/gender you prefer, the point of view and story lines are not much different. At times, it's pretty much identical. Same thing with games that let the player pick which class they prefer as in Baldur's Gate, Final Fantasy and Shadowrun (GENS version).