The key to whether it is self employment income is whether you do it to earn money. And free products count as money. So if you're writing reviews just to be writing reviews as a hobby then it shouldn't be counted as self-employment income (although it would still be taxable).
But if you're doing it for the free products then it's a side gig and it is a self-employed job. But - as you point out - self-employed jobs have expenses. In order to earn the 'money' from Amazon you may have a home office, which would be deductible. You definitely have a computer that may be deductible - at least for the percentage that it is used for writing reviews. Same with the internet necessary to earn this money. A hobby does not pay self-employment taxes but also can't deduct expenses.
As for which entity issues the 1099-K - in this instance Amazon is the payment processor for all of these independent sellers. As the payment processor they are definitely responsible for issuing the 1099-K.
The question of depreciating the items that you have been paid with is trickier. First, the items that you have received are essentially barter income. You've traded services for goods instead of cash. In order to be depreciated an asset has to have a life longer than a year and utility to the business depreciating it. So if you receive a box of tissues from writing a review there is no way you could justify depreciating it as an asset of the self-employed business that you are running. But a new laptop? I would definitely depreciate that as an asset.
@RichardH
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