Is there any relationship between grammar being ambiguous and the language itself?
There exist context-free languages which have no unambiguous grammar. Such languages are called "inherently ambiguous".

On the other hand, there is no such thing as a language which does not have an ambiguous grammar; any grammar can be made ambiguous by duplicating some or all of the non-terminals.

The fact that inherently ambiguous languages exist is interesting, but they rarely if ever show up in real-life parsing problems.