From a legal perspective, the writer of the review owns the copyright to it. You cannot copy and paste it elsewhere any more than you could post a chapter from Harry Potter on your blog. Unless, of course, you get permission.
If you want to use someone’s review for promotional material, all you have to do is ask. If they say “no,” then there’s nothing you can do, but that’s unlikely if you take the right approach.
A positive Amazon review is essentially a testimonial. As mentioned above, peer reviews are perceived as much more sincere than what the seller or spokespeople have to say, making testimonials a powerful marketing agent.
If someone posted a particularly positive, particularly eloquent review on your Amazon product page, you could use that for your own website, product packaging, press releases, email newsletters, social media — anywhere! But in order to use them, you need written permission from the author. Unfortunately, this is challenging due to Amazon's communication guidelines. You can comment on the review to ask for permission from the author, but the chances of the buyer seeing your request are slim, as they will not receive a notification.