It is illegal if it is a condition of a private flight that a particular passenger must be carried. Thus "Would you mind taking Fred with you" is fine, but "You must take Joe with you" is clearly not. Which would appear to be the case here
Really? Where is that stated? Suppose I offer to let you fly my aircraft if you take my sister with you, since she fancies a flight, and I have a cold. Is that illegal? Suppose a school says a PPL can do a repositioning flight if he gives someone a lift to wherever it's going. Is that illegal? No-one is being paid in either of these cases.
And if any of the above are illegal, could you please tell me where that rule is.