ELI5: Why are burns due to chemicals or intense cold also called “burns”? What do they have in common with the regular burns due to fire or heat?

In this regards,
this week, I held freezing metal for too long and have spots on my fingers that are pained and reddened. This frostbite is no different from a sunburn and will heal in a few days. Or this 'damage' and feeling is the same as if I were to get a mild sunburn from sitting outside.

There is no physiological difference between this and getting a sunburn, as it is minor and the body will fix it.

Acid/base damage is the same. If it is minor, then the body will recover easily from it. I've occasionally had small drops of concentrated acid/base on myself, once its been washed away, its fine with no lingering effects.
However, often the acid/base damage from malicious individuals is fairly intense, eyes will get exposed, and other body parts will be soaked for extended times. These are a bit scarier as the chemicals have time to penetrate the skin and do damage. Its these that cause the 2nd/3rd degree burns and are the disfiguring 'burns' that you're thinking of.

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