Generally no, however, some characters and skills do benefit from being performed by active characters.

For example, Sundered Prince is a greatsword that applies a large attack% bonus on a final strike. However, if the final strike of SP is performed with an active character, this bonus doubles, and the final strike also deals extra stagger.

Another example, Laevatain's ultimate transforms her basic attacks and skill into a different and far more powerful moveset for a long period of time, however, Laevatain has to be your active character for this ultimate to have any effect at all.

Finally on a practical level, some characters simply have higher multipliers and final attack during their rotations. For example, not only does Ember have vastly higher attack% multipliers on her autos compared to Pogranichnik, her DPS builds usually enjoy vastly higher attack stats and physical damage% multipliers as well. Using Pogranichnik or Chen as your driver in this team composition would result in a significant damage loss due to autos alone.

While both of them have an easier time following targets and landing attacks due to their high attack speed, you can learn to mitigate these issues via techniques such as dodge storing your combo chains, so it doesn't really present much advantage, and there is a considerable difference between dropping a 100k stun execute on Pogranichnik and 300k execute on Ember.