In English we use the single word "spicy" to refer to a broad range of flavors that all generally:
- involve a chemical sensation beyond simple taste/smell
- said sensation tends to feel "warm"
Some disagreement about how many unique kinds of spicy there actually are, but people typically agree on at least 5. Capsaicin (hot peppers), cinnamon, wasabi/horseradish/mustard, garlic/onion, black pepper.
Mint/menthol on the other hand have a chemical sensation that feels cool. I personally think ginger feels closer to menthol than to Capsaicin, so I tend to think of it as a cool flavor, but lots of people online lump it in with spicy flavors, so maybe I'm out of touch. Idk.