When an airplane is in a body of air which is moving across the ground, the airplane moves within that body of air with no knowledge that it is in a strong wind. Airspeed is unaffected, except for gusts or shear events. The airplane maintains it's normal airspeed within that body of air, even though that body of air is moving very quickly in relation to the ground.
This is evident in all phases of flight. For example, landing into a strong headwind does not change the approach speed required or the thrust required to obtain that approach speed. The only allowance is for gusty conditions, during which a gust will momentarily affect airspeed due to the inertia of the airplane. The heavier the airplane, the more time it takes a gust or shift to defeat the airplanes inertia.
The former doesn't make any sense, but the latter is what the GP was saying.
The poster I replied to was implying that due to the strong tailwind, they had to maintain an increased airspeed. That is incorrect. The airplane has no idea it is in a tailwind, and airspeed will be unchanged.
I think calling it a "common misconception" is also assuming a lot.
No, he wasn't. He explicitly said the opposite:
"higher ground speed to keep the same relative air speed."
The original was "relative air speed", which I would simply take as the way a person not familiar with aviation terminology would say "airspeed"--speed relative to the air.
This is further compounded when they say “they need to go faster”, as though the pilot needs to push the throttle harder. In actuality, the pilot doesn’t do anything differently to maintain their normal cruising speed.
The correct phrasing would have been the relative air speed and high wind speed combined, to result in a higher ground speed.