There are definitely signs for river and warrior. River is a combination of “Water” + “Street”. “Warrior” would basically be the same sign as “soldier”.
Some signs use the first letter as the handshape, but most don’t. ASL came from France, so some of the signs we use still have French origins… For example, “Search” uses a C handshape, because it’s based on the French word “Chercher”.
They were specifically talking about signs for the Amazon River or the legendary Amazon warriors, not the words “river” or “warrior”.
OK, I’m a bit slow sometimes… There would probably be signs in the local Sign Language for the Amazon River if you lived nearby, but here it would be a combination of spelling Amazon and signing River.
As far as Amazon Warrior, it wouldn’t have a special sign. You would probably Sign “woman + warrior + Greek + stories + long-time-ago”… or whatever you mean by it.
ASL isn’t word for word English… Although hearing people usually think that one sign = one word - which isn’t necessarily true.
The Deaf person who came up with the “Sign” for Amazon grew up not knowing how to Sign, so what he’s calling a “Sign” on their press releases, etc is really just him fingerspelling Amazon. It’s possible that he thinks it’s a Sign, but it isn’t.
It’d be like a hearing person saying they’ve invented a new word for Cat - and spelling out “C-A-T” - it’s not a new word, it’s just the word spelled out.
This whole thing seems like a good PR stunt.
Amazon should have found out what the deaf community signs for Amazon. I’m not sure if the same sign is used everywhere.
Deaf kids make up their own informal signs. The younger kids make up slang in sign language. Just like other kids make up slang their parents don’t understand. 
Most of my deaf friends have a sign they use to identify themselves. It’s used by all their friends. It’s similar to a nickname.
That’s not “hearing” people, it happens with every language pair. But that two languages don’t have identical structures (either both a single word, or both a multi-word expression, or the same type of word, or…) for a same concept does not mean the two cultures don’t have the concept.
BSL and ASL don’t have copulative verbs. English has one. Italian, Catalan and Spanish have two; which one is used depends on whether the union is permanent (or hoped to be so) or temporary (or hoped to be so). All have ways to differentiate “permanent state” from “temporary state”, they just use different structures to do so.