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Probably a software issue - the TV is reading the incoming SDR signal and treating it as Rec. 709 (the SDR standard) as opposed to Rec. 2020 (the HDR standard). Since the color primaries for Rec. 2020 are much wider, the values in the signal representing the "amount" of each color are smaller. If you then treat those numbers as in the SDR gamut, you get a washed out (less saturated) image.

This is also just a shot in the dark, but matches your observations pretty well.

Whoops, this should read "the TV is reading the incoming HDR signal". Just wanted to clarify.

I can almost guarantee that the issue is your Fire TV's input is set to a specific picture mode, whereas the built-in apps are using the color-inaccurate vivid mode.

Go to settings, all settings, program tuning and on the first option there press 11111 and a diagnostic screen will show HDMI settings.