Bodies re-use a lot of the same chemical messengers to do things, sort of like how you can use letters to write lots of different things. This is controlled both by release of the the messenger, and by the presence of receptors, things which receive the message–if the messenger is your postman, for example, a house can’t receive a letter if there is no mailbox.
One of the chemical messengers released in the “fight or flight” response does a LOT of things, but it gets flooded everywhere at once instead of well-being well-controlled. This may have some benefits for survival (for example, better cooling as you run away). It doesn’t matter how you smell if you can get away from a predator, for example.
It should be mentioned, however, that sweating is pretty rare in the animal kingdom. Primates and horses sweat all over their body, and some other animals sweat in select areas, but it’s likely that, in general, its benefit for survival really only applies to certain situations and life histories. Primates and horses both engage in behaviors that allow them to flee or hide from predators, and both had ancestors which survived in jungles and other hot areas where cooling and being able to hide in the forest (both by sight and by smell) had a bigger advantage for survival.
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