Not quite (I think): .NET Standard is an API definition, or rather a versioned set of them. ".NET Framework" (e.g 4.5) "implements" that definition, with some extra features to boot. ".NET Core" also implements it. It's a cut-down (though not very), cross-platform implementation of the .NET Standard.
Mono is another implementation of the standard.. cross-platform too, but this time compatible with the actual .NET Framework, i.e. 4.5.