Words are objects in a sense, although they are abstract, but there is no singular objective language in the same way that there is no objective gender. Both are intersubjective, they are interactiongs negotiated between subjects. There is no object that you can point to and call “language”.

And your argument could be applied to expressions of gender. A feminine dress is an object, and a beard is an object. These are gender signifiers, but that doesn’t make gender itself objective in any way. The analogy to language is very close. They are both sets of signifiers.