Also the problem has been resolved by contacting the support. The guy said, that the WiFi being shared is of that new standard, which is not supported by my device/Amazon Photos. However... I do not know... My device is less than 6 months old (but true, it is pretty chip, so maybe they used old technologies) and all other apps work... No idea.
I will try to describe here, what the guy did (unfortunately I saw not everything - he was doing pretty fast :D ).
Also he opened the settings of my router (192.168.1.1). From there he opened "Network Setting - Wireless":
Then he opened one of the tabs here:
There he changed the type of WiFi to be shared (if I remember it correctly, it was selected the bottom one, something with "802.11 ... mixed" in name, and he selected the one 1-2 lines above, also with "802.11 ... mixed" in name, but a bit differently).
The guy himself said, he deactivated AC (whatever it is :D).
Hopefully it will help somebody :)