I've resolved this through discovering that I was targeting 'Mixed Platforms' when handling the build configuration. Instead I need to look at the 'Any CPU' build configuration and turned it off there. Once I did this issue was resolved.
I was reading about parametrization, but that's not my case.
I am injecting connection string right via Azure, everything is working (my deployment problem was solved by deleting azure service and creating it again -.- :D) but I am hacking a bit the build in appVeyor - I am creating an empty file "ConnectionStrings.config" jut to tell AppVeyor that it's all right, the file exists:
(line in my appveyor.yml)