It clearly does not apply to Amazon affiliate payments only, but is asking for a specific affiliate program with the manufacturer being set up - if it were Amazon affiliates, there'd be no need for them to be involved.
The optics of asking for that and after being denied changing the recommendation (presumably to something where they do get affiliate payouts) are quite bad.
That's a problem for the Wirecutter with products not available on Amazon: their (IMHO reasonable) argument has been that since they get the payout from a Amazon and not the individual vendors, they have no incentive to recommend a product that's not the best, since they get paid for whatever they recommend. This only works as long as everything tested is on Amazon. As soon as they start creating individual agreements with manufacturers, it becomes a lot less obvious that they can be trusted to not optimize for the most profitable affiliate terms. (Maybe there'd be something about having a policy that all agreements have to be the same, but that's tricky)