If you attempt to purchase a kindle book from within the kindle app, you will not be able to. Nor will there be any messaging explaining how to make a purchase (because apple does not allow such messaging). You’ll just be mysteriously confused about why you can’t buy a book.
You can use your credits from your subscription, but you cannot pay for a new audiobook in the app.
It’s a relatively recent feature (past year), but it’s possible.
No idea why the price through the app was better than the price on the website. Maybe some kind of promotion? Weird.
I'm not surprised, since you can often buy a Kindle + Audible book for cheaper than buying the Audible book alone.
If you look at the Audible book (such as Wool https://www.audible.com/pd/Wool-Audiobook/B0BKR3Y6SP) the price is often higher than if you look at the Kindle book price, and adding the Audible narration (https://www.amazon.com/Wool-Hugh-Howey-ebook/dp/B088SY4GSD).
In the Wool example, the Audible book is $31.64, but the Kindle book with the audible add-on is $9.48 ($1.99 + $7.49). Even if the Kindle book was selling for "list price", it would still be cheaper at $27.48.
So, I'm guessing there's just a lot less margin headroom to eat a 30% cut on Kindle prices, but plenty to eat that on Audible's prices.
I don't see how Apple would prohibit me from visiting a website, and they've allowed use of content/subscriptions purchased elsewhere in iOS apps for a while now.
What has not been allowed (at least until today) is to directly link to that external website.