Please note that the pain scale is usually used to evaluate the efficacy of treatment. As long as the pain is responding to treatment, there need not be objectivity per se.
Was it a '7' on arrival? is it a '2' after a couple of doses of an analgesic? (if so, Great! Can we safely get it to a '1' or '0'? Did it spike to a '5' today? Why might that be? Is there something the team is missing that is causing the pain to be poorly controlled?)
This is the goal of the pain scale. For this reason, there is no need for objectivity.
Can pain be objectively measured, therefore reported?
The intensity of pain (as you have mentioned) is often left to the patient to describe on a scale of 1-10, or a visual analogue of faces. There is nothing objective about these methods, nor can there be, because pain is not objective; it is subjective.
Pain is subjective
Pain is a subjective experience; you cannot tell with certainty how much pain your fellow human is experiencing, which is why we ask people; they then can tell us. Pain relief (both physical and emotional) is a significant part of medicine, yet we still have "pain scales" for self-reported pain, one of the more common ones being the Wong-Baker Faces Pain Rating Scale:
To try to accurately assess pain (which is still subjective), a patient's scale should be interpreted by a caregiver using examples appropriate to that patient (If the person has had severe kidney stones, for example, the examiner can use that as a "10".)
...the worst pain I ever had was a kidney infection where I eventually passed out. If I take that as a 10, very few things even get to a 7 ;-) which is why I am asking. So you're basically saying physicians expect a high number much earlier than that?
No, we don't expect a lot of 8/9/10s on the scale. We hope it will be used exactly as you have described: in many cases, 10 is described by a caregiver as "the worst pain you ever had." If the patient has never had severe pain before (kidney stones, childbirth [for most], etc.) the pain has to be imagined.
Left to their own devices, a patient might look like a 6, but be reporting a 10. In this case, a nurse must try to ascertain their actual level of their pain. But still, it is their pain, and how it is felt differs from person to person, which is why no objective criteria can be assigned the pain scale.
The subjective experience of pain: Where expectations become reality