Yes, A difference in the accounts. You have a SmartThings account. Now, you have to go through Samsung account. And Samsung account signin is not compatible with Silk. And Fire doesn’t allow you to make anything other than Silk the default browser. So there’s no way to jump from the app signin to Samsung signin with Chrome.
I’ve done live chat twice with SmartThings support and exchanged emails with ST support. They are unable to help except to say officially that ST app does not support Kindle Fire. I’ve gotten unofficial guidance from all three conversations – but none of it was fruitful.
My best guess is that Samsung has no desire nor plan to work with Kindle Fire, probably because of contractual issues with Google.
It really does seem to be so simple as to just adjust the damn login process.