why can't I call this function from the code?

object Extensions {

    inline fun <reified T : Any> SharedPreferences.getObject(key: String): T? {
        return Gson().fromJson<T>(getString(key, null), T::class.java)
    }
}

can't call mySharedPrefs.getObject<Object>(KEY)

Well you should be able like this:

with (Extensions) {
  mySharedPrefs.getObject<Object>(KEY)
}

But perhaps it would be better to make it top-level function

Yes that works.

Unrelated, but you should avoid creating a new Gson instance every time you fetch something