[mavamaarten]

Fire devices just run a flavor of Android, without Google Play Services. Getting SSH on there is trivial, especially when rooted.

[iAMkenough]

I would like to install SSH and root a Fire Stick with my phone. Where should I start?

[Nextgrid]

Second result on Kagi: https://www.firesticktricks.com/install-apk-on-firestick-fro... ?

This would at least give you the ability to install an SSH or RDP client since those don't even require root.