'Objective' is a kind of antonym of 'subjective'. What it means in this context is that there are certain things which when measured in identical circumstances but at different times and places, will give the same answer. This is the nearest we can get to a test of 'reality' - that the outcome does not depend on the state of mind of the observer or idiosyncrasies of the measuring process.
It's the basis of physics. An objective property is not coordinate dependent, but measurements of it may be. But in the inertial frames of SR, all the local frames are the same so that is not the case here.