In that situation I would definitely identify myself as an EMT. What's the worst case scenario? They tell you they don't need your assistance? Even if there was a doctor on board he could be a psychiatrist or a dermatologist and therefore probably less qualified to treat an acutely ill or injured person than an EMT.
I called up my mom to ask about this because she's a flight attendant for United Airlines. She told me that all flight attendants are trained annually in AED, oxygen use, CPR, and first aid. There is an AED on every United plane, and I bet all of the other major carriers have AEDs on all of their planes too.
There are always at least two basic first aid kits on the planes, mainly for use by the flight attendants. Those kits just contain the first aid basics.
There is another kit that is kept in the cockpit. She said only doctors, paramedics, and RNs are allowed to use it because it does contain drugs like atropine, epi, glucagon, lidocaine, lasix, narcan, D50%, etc. The special advanced kit also contains equipment for IVs, urinary catheters, OPAs, a BP cuff and a stethoscope.
She said that there is oxygen in the cockpit, first class, at various points in the plane in overhead bins, and in the back.
She also told me that once a flight attendant starts using an AED on a person they are not allowed to turn that responsibility over to anyone else, not even if that person is a doctor.
She also said that CPR must be continued on someone regardless of their apparent condition until landing.
Now these are just relevant to United but I think it's extremely likely that most major airlines have very similar if not identical procedures and supplies available.
This company sells medical kits for commercial airlines.
The FAA requires that any plane of more than 19 seats must have the following equipment as a minimum:
-16 adhesive bandage compresses
-20 antiseptic swabs
-10 ammonia inhalants
-8 bandage compresses
-5 triangular bandages
-1 noninflatable arm splint
-1 noninflatable leg splint
-4 roller bandages
-2 rolls of adhesive tape
-1 pair of bandage scissors
-1 pair of protective nonpermeable gloves