Even given your second point, I would argue with the first, given that I think that globally, the company is more thought of when someone sees the word "Amazon" in 2020 than the rainforest.
That's not even the important point. Ownership of a domain name doesn't have to do with who is more of the owner of the name, nor whether the name is "etymologically pure", or whose use of the name is more popular. The entire right to own a domain (or TLD) is based solely upon ICANN's Uniform Dispute Resolution Process, and any established case law which exists in your particular jurisdiction. So this thing that the majority of the world's communication and commerce is dependent upon is adjudicated by a private entity in a single country that makes its own rules.
So it's not AMZN that wants to own the word, it's South American countries that want to do so, even though they have no more right to it than the company.
They had the word first. That is a compelling argument.
Imagine I start a company today called "America" (I'm in the UK). My business is wildly successful and grows to the point of being a global unicorn that can afford gTLDs. Should ICANN allow me to own .america? A lot of people would argue I shouldn't..
"The name 'Amazon' is said to arise from a battle Francisco de Orellana fought with a tribe of Tapuyas. The women of the tribe fought alongside the men, as was the custom among the tribe. Orellana derived the name Amazonas from the mythical Amazons of Asia described by Herodotus (see The Histories [4.110-116]) and Diodorus in Greek legends"
The point is that the word "amazon" (which is what is being bought as a gTLD) was not first used in South America, so they have no claim to it beyond "we use it too". Which isn't a good claim, otherwise they should've/would've been able to prevent Jeff Bezos from calling the company Amazon in the first place.
No, they used it first and Bezos named his company after the river. He could have chosen another name instead of using one that was already taken. He's the one that caused a name collision so he's the one that should suffer from it.
In English, "the Amazon" could also be a valid phrase apart from the river. For instance, you might have a course called "The Amazon in Ancient Greek and Modern Culture".
No, it would still be "The Amazons from Ancient Greece" or "The Greek Amazons" or something. "Amazon" singular in that context is only used as an adjective, such as "the Amazon Queen".
Quick edit: Ah wait, I think I see what you're doing with that phrase. It's shorthand for "The Amazon culture" or "Amazon depictions" or something, but it still comes off as strange-sounding to me.
Using "the X" to generalize about a group of people is likely to trigger offense if you're talking about a group of people who exist or is identified with today. It's not just the essence of stereotyping, but also kind of affected and pompous.
So, sure, it sounds "strange", but it's not new, and people use it when they want to make a generalization with a tone of authority. Probably one might expect to get away with it when talking about an abstract concept or archetype that exists independently of actual people.