Healey:
Actually that's the badge for the chief competitor in insurance and roadside rescue to RAC (the Royal Automobile Club). Your's is the Automobile Association (AA). I also had a membership to the AA (found the 80s version of that badge in the same box as my RAC one).
You joined these organisations and they both had their origins in the club culture. The RAC had a very fancy club with all the Mr. Patel features (I think they still have the club building). They were originally a toff's auto enthusiasts club, were one of the first racing sanctioning organisations and created a roadside assistance organisation patrolling all through the UK. At the same time the AA was doing similar. Both date back to the turn of the 20th century or thereabouts.
In the '80s the badges were square for both RAC and AA, and the display entitled you to some advantages when either would show up for roadside assistance, usually in some Ford Transit or Bedford van back then.
Yours looks like my father's; I think that design started in the 40s just after the war and was replaced by the square one in the late '60s or so. It's 'way cooler than mine were, but I still have his RAC badge (which is very pretty too) and an AA one like yours.
I think the old AA was the parent or the inspiration for the American version, American Automobile Association AAA.
I'll get some photos for you, just for fun. They're in the garage and its raining right now...
I'm back from a soggy trip to the garage:
Here's what the RAC emblem looked like in the '80s...

...and the AA emblem from the same period.

These are the old 50s and 60s era ones. They clip to the grill of your car.

I'm tickled we found the box they were all tossed in. I have no idea whether they still are in the same role in the UK today and, if they are, what the badges look like if they even still display such things.