Air Carrier Captain reported the illegal transport of passenger's personal Oxygen bottle. Captain stated the cause of the Hazmat violation was due to an inconsistent Hazmat policy agreement with Regional airline partners.

Narrative

Upon the arrival of my Aircraft X (our last flight of the day) at ZZZ; our last passenger was a blind woman who needed a wheel chair; after helping her to her chair I noticed she had a 'thermos' (it seemed more 'complicated' than a thermos) in a neoprene sleeve. On the way to the hotel; I asked my FAs (Flight Attendant) if that item was actually a thermos or something else? #2 FA responded that it was a small Oxygen bottle. I was surprised. I mentioned that Oxygen is not allowed in the cabin; and my FA told that it was allowed and- that passengers do it all the time - I was very surprised. I told him how dangerous it was and that he should tell me the next time anyone brought Oxygen on board our aircraft again. That night; in my hotel room; I looked to see if there was any change in the regulations and I did not see any changes from the Hazmat training and Hazmat Information in FOM -nor did it make sense for passengers to bring Oxygen on board an aircraft. The next day we dead-headed back to ZZZ1 and as I got off the airplane I asked my #2 FA (again) if he knew the difference between an Oxygen bottle and a POC (Portable Oxygen Concentrator)? He was very sure he knew the difference and he said he was sure that it was OK for passengers to bring Oxygen on airplanes he said 'they have been doing it for years' and while he was relating that to me he said 'look behind you' and there; in my amazement; I saw a passenger pulling a large Oxygen Bottle (I would guess 680 liters) on wheels toward the upper B gates (inside the secure area of the terminal) on his way to a flight. I was stunned. (I would like to mention that my FA was extremely respectful in our conversation and at no time did we argue; he agreed with me; yet was honest enough to tell me the truth of what he has seen and I was appreciative of his candor) At this point; I started to realize the problem was larger than I thought. Next; I called ZZZ1 Operations Tower; and related what I had seen and the Duty Manager was just as surprised as I was (he even looked up the passenger in [the] Regulations and found no special remarks). After I finished the call; I proceeded to my outbound flight. I spoke to both of my FA's and told them that Passengers bringing Oxygen on board our aircraft is Hazmat discrepancy and that it is very dangerous and that if they see Passengers bringing Oxygen on board -that they should call me immediately. A little bit later; while we were getting ready to close out and depart; we had a minor problem with a passenger which was resolved quickly by a passenger service Supervisor; as the Supervisor was departing the aircraft; I asked him about Passengers boarding our aircraft with Oxygen and he told me: That it is not allowed on our aircraft; HOWEVER Company allows the Captain to decide if it's OK to have Passengers bring Oxygen on board their aircraft. I believe; that what is taking place is a chronic reportable discrepancy of bringing prohibited substance on our aircraft and that; by Company accepting passengers to bring O2; our Gate Agents have gotten confused as to which airline can allows this; and possibly have become confused at the correct procedure (I have seen this; in many instances; when multiple airlines use the same passenger service and Ramp agents (at HUBS : ZZZ1 & ZZZ2 particularly) and Outstations as well the differences in airline protocol lead to daily discrepancies due to agent confusion. I would like to make a note here: Oxygen is very dangerous; that combined with up to 5 liters of 70% alcohol ; Passengers can bring onto and aircraft and a Lithium Ion battery would make a very powerful bomb and oxygen can simply used as an accelerant for an interior aircraft fire and finally; what is actually in these compressed cylinders? Many poisonous gasses as colorless and odorous and could seem like oxygen. How are the Passengers getting these compressed gas containers through TSA? And our Ticket Counters and Gates??? Does Company1 havean allowance for allowing Passenger to bring on Oxygen? Wouldn't they need a spectrometer to see if it was not Poison? Why does Company2 Hazmat Training Program tell us that Passengers cannot bring Oxygen on board and yet Company is different? Cause - INADEQUATE Training and SOP implementation. Lack of Streamlined operational SOPs with Company -causing ZZZ1 Passenger service to allow Passengers to board Company2 aircraft with Oxygen. Company lack of oversight and disregard for HAZMAT procedures and safety of their passengers and crew.Suggestions - Examine Company airlines HAZMAT program -how can they allow passengers to bring aboard Compressed gas in cylinders?. How are these Passengers getting Compressed gas bottles into the Gate/ Terminal area and on board aircraft? Examine Ticket counter and TSA SOPs. Unify HAZMAT operating procedures between regional airlines (that utilize the same passenger service agents and ramp agents) to avoid confusion. Investigate why Company has a different HAZMAT procedures versus Company1. Investigate if CFR's are being violated.