Is it possible to get a windows-version from a powershell command on Windows Server 2016 Nano?

You could try the following, I've not got a nano server to try it out on. Drop the select if it gets you something else and see if what you want is stored under a different property in Server 2016 Nano

Get-CIMInstance -ClassName Win32_OperatingSystem -Property * | select caption

When tested on a real Nano instance the -session parameter was not needed, but if you need it at some future date, here's the variant with -session:

$cuser = "Your username"
$cservername = "Your Servername"
$csession = New-CimSession –Credential $cuser –ComputerName $cservername
Get-CIMInstance –session $csession -ClassName Win32_OperatingSystem -Property * | select caption

This is just an extension on your edit, but cleans up the output, by getting only ProductName

$(Get-ItemProperty -Path 'Registry::HKEY_LOCAL_MACHINE\Software\Microsoft\Windows NT\CurrentVersion' ProductName).ProductName

The Microsoft way aka the Cert way is to use the Get-WindowsEdition -Online

additional information on the command and all its options can be found at Here!