A very little known fact about Amazon’s Echo and Echo Dot is that they actually run on Android. I wouldn’t necessarily call it Fire OS, which is Amazon’s variant of Android, but around when the second generation Echo Dot was released, Amazon switched from a generic Linux based operating system to one based on Android. Despite not having a visual interface, Echo devices run standard APK apps to achieve their functionality. The Echo Show will almost certainly be running Android, and will probably have more in common with Fire OS on the Fire TVs and Fire tablets than anything else.
Since Echo devices run a variant of Android, there’s really no reason to think the Echo Show will run anything other than Android, and it’ll probably share many OS components with the Fire TV and Fire tablets. The switch to Android for Echo devices was probably an effort to converge Alexa software development to a single platform, since the Fire TV eventually gained Alexa capabilities, as did Fire tablets.
While the Echo Show’s interface and capabilities will likely be achieved through standard APK Android apps, just like the Fire TV, don’t expect Amazon to allow users to install anything resembling an app on the device. The Echo Show’s “app” ecosystem will be entirely based on Alexa Skills instead of the Amazon appstore.
It’s abundantly clear that, despite having a built-in touchscreen, Amazon wants voice to be the primary interaction method for the Echo Touch. In Amazon’s nearly 5 minute long introductory video for the Echo Show, not once do they show someone touching the screen. They don’t expect, or probably want, people to use the device for hours on end as if it were a tablet.
Despite how Amazon wants the Echo Show to be used, running Android and having a touchscreen is going to make it a prime target for modding. I wouldn’t be surprised if, within weeks or even days of its release, the modding community figures out a way to install/sideload 3rd-party apps on the Echo Show which allow it to double as a standard tablet. If the device eventually gets rooted, which is almost inevitable, the sky’s the limit.