I think it's fine for a champion to recycle some mechanics from previous kits (and, in fact, it's impossible for them not to), but at the end of the day I think a champion needs to play in a manner that is sufficiently distinct from any other champion to be worth the extra champion in the game. A champion needs to have at least one thing going for them that is completely, unambiguously unique, and that unlocks a completely new way for League to be played. If that is missing, then you might as well just take the champion you tried to copy and enable some alternate build for them.
With that said, uniqueness does not equal quality, and I very much agree with the OP that uniqueness purely for uniqueness's sake can be harmful to the game if done wrong. At the end of the day, it's important for a champion to be a unique, but it is equally important for that uniqueness to be fun, and so for everyone involved. This doesn't mean that champions can't be allowed to have ridiculously impressive effects (in fact, that can be a great thing), but this does mean that the sheer wow factor of those effects should not stop them from getting cut if they turn out to be unhealthy. This is why stuff like "let's give this champion infinite dashes", or "let's give this champion a chance to cast Redemption in lane" needs a lot more consideration beyond just how wacky that kind of thing sounds, since in practice those mechanics have proven to be unhealthy to the point of likely being irredeemable.
One issue I've taken with some more recent designs is that some "unique" things are only unique because they contradict preexisting design rules: Zoe, for example, initially stood out for being designed to literally one-shot people, and while that's interesting from a design meta point of view, in practice it's not fun to deal with, and sometimes feels unfair even to the Zoe player, and not in a good way. Similarly, Ornn's greatest claim to fame upon release was that his ability to buy items outside the shop sounded super broken, but in the end that's not something most players care about all that much, at least not when compared to his raw damage and CC in the past few months. By contrast, someone like Bard stands out not because he "breaks the meta" by abandoning his laner, but because his entire character and playstyle are so unorthodox that it really crafts this fully-fledged experience of going on a magical journey every match.
At the end of the day, a champion exists to provide a gameplay and thematic fantasy, one that should stand out from every other. This means a champion needs to play in a unique manner, but also needs to respect every other pillar of gameplay, namely them being genuinely fun to play and to play against. A champion that's healthy, but not unique, merely takes up unnecessary space, but a champion that is unique, yet unhealthy, actively harms the game whenever they're present. As such, while I am very grateful that Riot has been pushing the envelope and going much bolder on designs, I also think that should never conflict with the need to make those designs properly interactive and beneficial to every player's enjoyment, not just that of one player at a time.
