I suspect there may be a misunderstanding, on his or your part, about what "affiliation" means.
It may be that rather than parsing it as "a declaration of groups you are directly affiliated with", he's parsing it as meaning "a declaration of any directly or indirectly affiliated group that may have had an influence on your work". In that case, yes, it's fair for him to say that your advisor's affiliations may influence your work. If, say, his research was primarily funded by Nestle, nobody should take your word if you say "Nestle is super moral about All The Things". But the inverse is not true; your affiliations won't affect his nearly as much. That is: distrust and influence-taint are commutative, but only downhill.
That interpretation seems a reasonable and even likely one for someone to mistakenly come up with.
Still... any time a superior asks you to do something that might be unethical, you should have the request in writing, and your subsequent objection to it in writing.
Yes, getting it in writing can be awkward, and objecting can be awkward, but I find expressing uncertainty and asking for clarification to generally be the most tactful way of doing it. So in a situation like this, I'd ask for further clarification in email. In my writing style, I'd say something like:
I'm not sure I understand what you mean by "the student's affiliation
should follow the advisor's lab's affiliation". I'm not sure which of
the following it means, if any - could you clarify?
- You require me to lie about having this affiliation in order to publish (was my first knee-jerk interpretation, but honestly seems least likely: I admit it'd be completely out of character for you);
- You interpret "affiliation" as a declaration of potential influence, rather than one of qualification (I strongly disagree with this interpretation, and in this case would like us to clarify this with the editor before continuing);
- You think I already have this affiliation (I'm afraid I do not);
- You require me to pay to acquire the missing affiliation, in order to publish (I do not intend to);
- You require me to get it, but are willing to sponsor me to do so, in order to publish; or
- Something else (this seems most likely, as I strongly feel I may have completely misunderstood - apologies in advance for that!).
[Obviously, write that in a way that would be aimed at your supervisor and your own writing style.]
But the point is, in asking a question like that, we'd be putting the onus for the "misunderstanding" on ourselves, allowing him to gracefully "clarify", even by entirely reversing his decision. We gave him 100% of the benefit of the doubt, and covered his ass well while keeping it documented. You've gracefully resolved it, without harming him or burning bridges.
In the unlikely case that he continues to openly insist that you lie, and declines to let you get clarification from the editor... well, what you do with his written response then is up to you.
A fairly low-friction path might then be to go on over his head to ask for clarification from his superiors, or from the journal's editor, stating in either case that you suspect that he might have misunderstood some instruction they gave him, and ask whether they could offer you both a clarifying correction.
However softly you couch it, though, this could have some really big repercussions from him, if his publishing history is long and filled with such knowingly-deliberate falsehoods. Retraction of papers, blacklisting, etc.
But honestly, that's fair. Because, if he is (and you are!) willing to knowingly and deliberately lie about something as significant as affiliation, then why should any other part of your paper be trusted, or any future paper?