The tablo is connected to the same router that the android devices are connected to.
However the tablo is wired to one of the LAN ports on the router (which is configured as a wifi access point in bridge mode) while the android devices are connected via wifi.
Since the router is in bridge mode, the same IP subnet is used for both the LAN ports and the wifi connected clients.
It is definitely not a connectivity problem between the android devices and the tablo. The easy way to prove that is to open my.tablotv.com in the silk browser (on the Kindle Fire) which is able to access the tablo!
The very same tablet can talk to the tablo when accessing my.tablotv.com in the browser but is unable to access the tablo when running the Fire/Android tablo app.
This should be sufficient proof for a software defect in the tablo app. My guess is that the developers ignored the possibility of a bridged wireless - wired network and wrongly assume that any wireless device would only ever connect to a tablo that is also connected wireless. They are probably looking for the Tablo_#### SSID which is of course not present when the tablo is wired!
I sincerely hope that this problem gets fixed (reversed to the previously working behavior) before the same bug gets ported to other tablo apps (especially the roku one)!
The Kindle Fire is running Fire OS 220.127.116.11 (it is a 5th generation 2015 Kindle Fire), but I don't think that has anything to do with the reason why the tablo app no longer works. After all the app worked before and the tablet can still access the tablo from the browser (just not with the app).