It is against the rules. If you are an e-commerce company (i.e. a competitor) or you operate a shopping comparison service, you are not allowed to use the API at all.
That appears to be specific to their advertising API, not the affiliate program. The affiliate program policies[0] say:
> if you choose to display prices for any Product on your Site in any “comparison” format (including through the use of any price-comparison tool or engine) together with prices for the same or similar products offered through any web site or other means other than an Amazon Site, you must display both the lowest “new” price and, if we provide it to you, the lowest “used” price at which the Product is available on the Amazon Site.
... which implies that it's not forbidden, but I can't blame someone for not wanting to risk it.
> if you choose to display prices for any Product on your Site in any “comparison” format (including through the use of any price-comparison tool or engine) together with prices for the same or similar products offered through any web site or other means other than an Amazon Site, you must display both the lowest “new” price and, if we provide it to you, the lowest “used” price at which the Product is available on the Amazon Site.
... which implies that it's not forbidden, but I can't blame someone for not wanting to risk it.
[0] https://affiliate-program.amazon.com/help/operating/policies