I think it's harsh to call it extreme paranoia. I've heard that one of the reasons Amazon won't use Slack for its chat is because it isn't self-hosted. I'm sure many major companies won't use Gmail. I don't think Slack or Google have done anything that suggests they'd mine their platforms for corporate espionage. Slack isn't even in the same industry as Amazon. Is Amazon paranoid?
According to the article, Walmart seems hesitant to support so much profit running to Amazon. While Amazon's retail business runs on razor thin margins, their cloud business reaps enormous profits. That means that the cloud business is helping Amazon put a lot of pressure on Walmart. Without it, would they be able to eat such thin to negative margins in the retail business or would they need to raise prices and make Walmart's offerings look better for consumers?
In the article, they note that AWS was 89% of Amazon's operating income. It's likely that Amazon's thin retail margins would bother investors if AWS wasn't raking in money. It's also likely that AWS margins are helping Amazon be more aggressive in retail.
Still, I don't think it would be more paranoid than most companies to avoid using a competitor's infrastructure at that size. Amazon will prefer self-hosing email rather than using Gmail regardless of assurances.
It's not even just about trusting that the competitor wouldn't actively decide to peek into your data. It's trusting that no one in that company would take a peek even though that the competitive data could give them a huge career boost. Let's say Amazon hosts their email with Gmail. You're a struggling middle-manager at Google Cloud who has seen some failures. Maybe peek at Amazon's email and see what their cloud plans might be and when you're able to get out in front of Amazon you amaze your colleagues - maybe you have to bribe someone with access.* It's not just trusting that Google doesn't want to do this.
*Now, I'm sure that Google has safeguards to make sure that employees can't do this (or it's at least hard), but the point is that you have to trust 1) the company, 2) the employees, and 3) procedures the company has put in place to prevent nefarious data access.