That's not right is it? At least, not logical induction?
Supposing the second premise were 'Lily is female'; the answer to 'What gender is Greg' should obviously not be 'female'.
Inductive reasoning (induction) guesses the general case from particular instances.
I presume that you are thinking about deductive reasoning (deduction), which guesses the particular instance from the general case.
Finally, abductive reasoning is usually the goal instead of simple induction, and technically what the neural networks do: compute the simplest general case that best explains particular instances they are trained with.
Using their flawed logic, since Lily and Greg are both swans, then Bernhard must be a swan, so Greg could be white or green.
You're right about the answer not being correct, and not for the reason I gave. The test was a butchered version of the black swan problem. It shows that you can't generalize given prior examples. Even if you've seen a million swans, and they were all white, you haven't proven that all swans are white.
I haven't seen the show, so I don't know if they were intentionally going for that, or they screwed it up and honestly thought it was a proper example of inductive reasoning.
But induction is more like learning about the world by observing it, and making probable guesses. So it may be correct for a machine to observe that if one swan is white, the best guess for the color of another swan (in the absence of other info) is white.
I'm not familiar with that definition of induction, but I haven't studied ML at all.