Adding too much uniqueness hurts balance

The game is balanced fairly well around the fact there there are classes and sub-classes of champions with similar playstyles and strengths. Many can be played with the same basic goal, and the same basic weaknesses.

When champions are changed or introduced who have unique strengths that others in their class simply lack, they become the best option. The biggest offender of this trend was Kassadin, waaaay back when he was the most mobile champion and could silence opponents. Ivern, with his entire kit, is another good example. Though it could be said that there isn't a similar supportive jungler at all.

Some unique strengths add diversity in a healthy way (ex. Tristana's E passive compared to Ashe's slow passive, both provide differing strengths but neither makes the champion a drastically better pick). But adding too many onto one champion makes it impossible to pick a comparable opponent. (back to the Ivern example, there is no other champion in the game with his passive/w/ult, and no other jungler with his build choice)

In the end, making it so that a champion isn't a catch-all for every role is a good choice to make. But making it so champions intended to fit a similar role all fall into different niches or have strengths others can't replace is not.