Medical/clinical addiction normally revolves around hard-to-control cravings (can't stop thinking about it, might get aggressive if prevented from getting it), loss of control, etc. Dependence is essentially making your body used to something, and then having to consume it to feel normal, or feel like shit when you stop consuming it. While addiction and dependence are similar, addiction is far more brutal and hard to quit. By no means coffee is as hard to skip as cigarette, to anyone. If your reaction is "wow, coffee makes my head hurt, I don't want it", that's pretty opposite of addictive substance.
https://www.science.org/content/article/coffee-cravers-are-not-addicts

I understand, but coffee is amongst the most consumed things in the world, yet headaches from it is something I don't really hear around, so it's not completely wild or out of the blue to suggest a check-up. Something similar was happening to me with energy drinks 10+ years ago, which was related to my blood pressure.