Answered by Steven Fowkes:
Copper is antibacterial and anti-fungal. So is silver. Many metals (and metal salts) are potentially antimicrobial. Silver and copper are fairly benign to human skin, so these are widely used; you wouldn’t want arsenic or antimony in your socks.
The attachment of the metal nano- or micro-particles to the fabric is critical. Merely soaking your socks in copper sulfate will not sustain the anti-odor effect; it will wash out when you launder the socks. Although you could spray a copper solution on your feet or inside-out socks before putting them on.
re: Couldn’t a person develop an allergy to copper after continuous exposure?
Not an allergy, directly, but certainly an intolerance or adverse reaction. For example, copper and zinc are to a significant extent antagonistic, so copper could induce a zinc problem. Copper is also one of three minerals that are sequestered (stored in the liver and in blood proteins) in response to infection, so repleting a type-II copper deficiency might aggravate an infection that is copper limited (i.e., copper driven). Copper toxicity to microbes causes them to induce chelation defenses that could make the excreted copper more absorbable through the skin. There are many ways copper toxicity might manifest.
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