Of course there is a certain degree of balance in single player games too, but because of the nature of single player games, there usually only a very little effort made to ensure all characters are roughly equally useful thorough the game, and huge imbalances/borderline exploits are rarely ever patched, partially because patching pre-PS360 era was pretty hard to do, but that just means you need to spend more time playtesting.

Some examples are:
Certain Enemy skills that are extremely powerful and obtainable early in the game (hi2u Beta, aqualung, white wind)
Certain limit breaks/overdrives (Duel vs Selphie's Slots, Omnislash vs Tifa's slots, attack reels vs whatever that thing auron had.)
Certain stat boost possibilities (speed junction in FF8), and the ability to gain extreme stat boosts in general early in that game
Maria in Star ocean 3, Reimi in SO4

The reason these imbalances aren't a big deal is that ultimately, single player games are human vs CPU, and the CPU is never going to complain about being curb-stomped (well not until we have true AI).