I found the cause - Anvil App Server uses Embedded Posgres database which is build (for Windows) using Visual Studio 2013 and thus needs Microsoft Visual C++ Redistributable Packages for Visual Studio 2013. After installing it, reboot the machine and Anvil App Server works as expected.
There might be other issues with Embedded Posgres - you can pin down the problem with running initdb.exe, -A, trust, -U, postgres, -D, .anvil-data\db, -E, UTF-8 within
.anvil-data\db-bin\PG-<…>\bin on command prompt and observe the behaviour.