Just tried this player. And to answer my own question: no, it does not have Amazon support. It does not have access to any "app store".
Compared to my old C5500 it has some additional nice options when playing streamed media files (like subtitles selection). Also, it is capable to play MKV videos with PCM audio stream (which is something C5500 did not support). Yet, for reasons unknown it gets stuck at endless "Loading..." screen for half of my MKV files. The old 5500 plays those very same files without a problem.
Also, out of the box, it was incabable of seeking inside MKV videos, but a software update fixed that issue (but not the infinite "Loading..." one).
So, in general, due to the aforementioned major problem, this product is virtually unusable as a streaming media player.
P.S. I was actually looking for F5900 originally. I wonder if F5900 is better? Will try it...