That would end the trend of pokemon outclassing each other.
I.E. Wigglytuff can perform almost anything a Clefable can, but do to stats and ability, Clefable outclasses Wigglytuff
But if Wigglytuff had a Signature move that was more ueful then that of Clefable's, then it would provide more incentives to use Wigglytuff.
Also a side note: I was pretty let down back in the day, when I realized that Jigglypuff wasn't the only pokemon capable of performing the move Sing. As a 7 yo who watched Pokemon back in the day, it wasn't clear to me why Jigglypuff was the only pokemon shown to be able to Sing, yet wasn't the only pokemon capable of learning the move.
Sing as Jigglypuff's signature (and evolutionary line exclusive ) move FTW!