Medical grade is the same but often has the moisture content increased to make it easier on the patient.
British Oxygen would definitely differ, bose x. I went through all this on the phone. The gas is the same, only the paperwork differs. No moisture is added.
Incidentally the rental for the biggest bottle from BOC is about £90/year, and a refill (via a swap-out) is about £20. Any welding outlet will be delighted to sign you up; the process is smooth and well organised. The payment is done by direct debit. That bottle lives in my garage (strapped to the wall so it can't fall over) and lasts me a year or two; in fact I end up swapping the bottle for a new one before the outlet pressure falls below the 150 bar destination pressure.
I have tried countless scuba shops. One has to go in under false pretences, and the success depends on how thick the assistant is. If he is clever (not common) he will suss that you are not a diver and won't do it. If he is really thick, we will just do it. If he is halfway, well it's a matter of whether you feel lucky or whether he feels sorry for you. I gave up on the scuba shop scene ages ago - wasted so much time driving around and begging.
The last time was in Greece when we spent a whole day driving around in taxis, being passed from scuba shop to scuba shop and eventually ending up referred to a man who owned a huge oxygen factory and they did it.
It's a waste of time doing it that way.
The safety precautions for oxygen handling are basic and easy to follow. Use the right equipment, the right materials, no grease, and make sure everything is tight. And don't sit on top of the bottle when you are filling it
BTW I have an Aerox aviation oxygen bottle right in front of me. The green band goes to 2250psi which is ~ 155 bar, not 137.
Obviously somebody with a helpful scuba shop nearby and who never flies very far will be in a different situation.