In my own attempts to find a solution for FireTV, I ran into this thread today. First of all, I can tell you that if you install the APK, it will work with any Amazon official App that can play music in the background (e.g. Amazon Music Player or Pandora, for example). Just set it to run your music and then go back to the home menu. As long as the music is still playing, you can then start ProjectM (either via the developer "manage applications" or with something like Firestarter installed from a secondary home menu). You'll then have ProjectM visuals with the music that was playing. The artist/song information even overlays over the ProjectM visuals here. For those applications/players, it works fine, although the controls/preferences really don't function without touch control (i.e. it's expecting to run on a phone or tablet).
I have had NO luck, however getting Kodi to work with the ProjectM stand-alone. You can set Kodi to launch an Android App (including ProjectM) from the favorites menu, but as soon as I do, the music stops (Kodi appears to quit in the background) and when I back out of ProjectM, Kodi restarts. That sort of launch might be useful for running completely independent programs, but it's not helpful to multitask and I have no idea what the problem is. I found a YouTube video that shows it working properly (it's using the Aeon Nox skin) but when I tried it, I could get ProjectM in the menu with Aeon Nox, but it did the same thing as the favorites menu; it quit when it launched ProjectM.
I have no idea who originally worked on ProjectM for x86, etc. and if they simply have no interest in the ARM version and/or if the guys that made the ARM version simply have no interest in Kodi (XBMC) any longer since it is for sale (for full functionality) on Google Play and what not. I did find a nice iTunes plugin today, though (
https://github.com/revmischa/projectm/releases) that I didn't know about in my effort to find more information.
I've also just attempted to contact the developers of ProjectM directly (at least their web site) to see if they can provide some answers. They are selling the App (slightly crippled for free) for the Android platform on Google Play. I wouldn't mind paying to have the functionality back in Kodi even if it's just a link-launched App. Some of us aren't coders and just want a working solution even if we have to pay for it. In other words, if they can tell me how to get Kodi to keep playing music when I launch their player, I'd gladly buy a copy of a pay version as I don't expect to see it supported here ever (or any other feature I might like to see based on past experience of seeing minor requests by people go ignored for over 8 years now).
Personally, I've sorted just about every other Kodi problem now on FireTV with at least a work-around except ProjectM (I've set it to just show a photo slideshow instead on a screensaver timer which is better than nothing).
THEATER: 11.1.10 Atmos, Epson 3100 3D Projector, DaLite 92" screen, Mixed Dialog Lift - PSB Speakers; Sources: PS4, LG UP875 UHD, Nvidia Shield (KODI), ATV4K, Zidoo X9S (ZDMC), LD, GameCube