My experience is that it is possible to have an app on your old fire tablet, but that app does not appear on the list of "My Apps", when you set up a new one...
In my case I am pretty sure it happened because that app is no longer on sale, but in that case you can backup from tablet 1, sideload to tablet 2 and it will install OK. it could also happen for " compatibility " reasons but it's more likely to be because an app is no longer sold.
I wonder if Kindle books are also like that. It is rarer for a book to go off-sale than an app so maybe harder to find an example
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