The better way to phrase it is: "Are there objective truths about language?" -- this question is parallel to the question of moral realism: "Are there objective moral truths"?
One way to interpret moral subjectivity is that it implies that "what people consider to be moral, is moral.". My analog for language is that "what constitutes language is just how people speak". This would be subjective in the sense that if everyone started behaving differently then language would change -- language is just what people do when they communicate.
On the other hand, linguistic realism would imply that there are facts or features of language that exist independent of anyone's actual speech -- maybe something like Plato's forms: there is Platonic Language, and then human languages are its shadows.
I know this is probably not settled, but I'm not even sure of who has addressed it, so, mainly I'm looking of indications of the main schools of thought, and outlines of their rationales. I believe that I need to be better acquainted with this question, before examining theories of meaning, sincebecause how I interpret andany particular theory of meaning will depend on the disposition of this question.