In this Episode, the anime Karneval has reached its climax, when the 'heroes' fight with their 'foes' in a final battle. In this instance, the mentioned Jiki and Nai are part of the 'heroes' group, and the mechanical sheep are part of their 'army'. The scene starts with one of these mechanical sheep 'dying'.
Nai: Mr. Sheep! *tries to reach for the sheep*...Mr Sheep! Mr Sheep!
Jiki: The sheep are just mere machines. A new one can always be made.
Nai: The sheep and I played together...
Jiki: *looks up at other sheep fighting around them* You wouldn't know if that's the same sheep right? There's so many of them, and every one of them looks the same.
Nai: But Jiki-kun...Out of those I've met so far...None of them had been the same. *scene changes to Nai viewing flowers* Mr. Flower and the other Mr. Flower...They were different.
Nai's reaction emphasises that no matter how similar or identical an individual can be, they are never completely the same. But this doesn't just stop at the corny fact: "Be yourself because everyone else is taken". No. What the dialogue is really emphasising is the sad reality of the ever increasing loss of individuality. In this modern age, we are living in a world swamped by stereotypes. Even life itself is a stereotype: everyone is expected to get an education of sorts, for most that's up to tertiary level, and get a job. For some, this gets even more specific, from the hobbies you pursue (ever forced to play an instrument?) to the type of profession 'you' aim for.
And you know, even if the above argument doesn't make any logical sense at all (in which, upon re-reading, I can see your point), the most thought-provoking point is this: Jiki's reaction that every single sheep are identical, so you can never really tell who's who". What's so sad about this is that in this world, so many people never really emotionally invest time and effort into really getting to know people for who they are. As a result, many people are just automatically lumped together into a certain category, a group with identical characteristics.
And that is the saddest part of the story, and why this dialogue was so emotionally received.