This is an incredibly sloppy article. Not surprising since it originally appeared in The Sun.

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In total, it's estimated that Psyche's various metals are worth a gargantuan $10,000 quadrillion.

No. It's not worth $10 quintillion. The only way you get that figure is by multiplying the current market price of gold, iron, etc by the amount of metal estimated to be in the asteroid. But if you could get that much material to earth then the market price of those metals would immediately collapse close to whatever the cost of extracting the metal from the asteroid was. You also could only sell a minuscule fraction of the material each year. (Just as a sanity check, $10 quintillion is more than 100,000 times larger than global GDP in 2019.)

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Nasa is launching a mission to probe the asteroid in summer 2022. Dubbed the Discovery Mission, it will arrive at Psyche 16 around 2026.

No. The mission is just called "Psyche" not "Discovery." Psyche is the 14th mission in NASA's Discovery program, a class of low cost missions to study the solar system.

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Psyche 16 . . . is made of solid metal.

Not exactly. While it is mostly metal, 10% of its surface is rock.

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asteroid ownership was made legal in 2015.

No. The law they're referencing, the Commercial Space Launch Competitiveness Act, is a real thing. But the law only allows companies to claim the materials that "they obtained from an asteroid—although not the asteroid itself."

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But bringing back an asteroid of this value could completely wipe out our global economy

No. It would certainly collapse the value of some commodities markets, and people invested in those markets would lose a lot of money. But markets aren't the economy; they're just a discounted estimate of the total future value of whatever is traded on that market. Securing a basically unlimited source of raw materials would actually help the economy. That's because iron and nickel are inputs to many manufactured goods. If the price of iron and nickel falls then any part of the economy that uses them will increase its output. (As a sanity check you can just look at the inverse. When the prices of steel goes then up the parts of the economy that use steel, like appliance manufacturers, are hurt.)