Why do my hands/palms smell like metal? (read the details)

I would be suspicious not of your hands, but your sensory perception. That is something which can be checked. Perhaps your sniffer is getting a confused reaction to the scent of your soap.
Has your water been tested?
It might be a good idea to mention this to your doctor.

I should go to a doctor? Makes sense.

You can smell your water and soap when it’s not on your hands and see if it smells like metal.

Is it just your hands? Or, your arm too? Do you have a metal taste in your mouth?

Did you start a new medication? Are your lips, tongue, and eye conjunctiva, naturally very red?

Go to the doctor. Maybe you have some sort of metal poisoning going on!

@JLeslie It’s only my palms, and it happens ocasionally, then stops on it’s own.
I don’t use any medicines, btw.

Do you ever encounter metal doorknobs? (At least brass… I’m not sure if stainless steel leaves any metal-odor). Maybe in your home, or maybe in public locations you visit? ... Public locations would actually help explain the “sometimes” quality of the odor, if this is what it is… Just a wild guess, though. I don’t really have any idea.

When I searched online I happened to run into a previous Fluther Q… no one had any answers or solutions there, either, but if you’re interested maybe it’ll inspire some new idea? Not sure… https://www.fluther.com/149339/help-me-get-rid-of-metal-smelling-hands/

@Soubresaut Thanks, that was interesting to read.

Doorknobs sound like a good guess.