[Suncho]

> It can't be just a number on a computer in a bank, right?

It's two numbers in computers. It's a number in the computer of the bank that issued the money, and it's a corresponding number in the computer of whoever is holding that money as an asset.

> Otherwise some Russian bank could just increase that number to whatever they like. And say "Look, we own 100 Trillion USD. Now let's go shopping."

It's not possible for a Russian bank to pretend they're holding more USD assets than they really are. The issuing bank knows how much USD is in Russian bank's deposit account. The Russian bank can issue its own USD liabilities (i.e. deposits) which are IOUs for the USD reserves that the bank holds on the asset side of its balance sheet. If the bank issues too many USD liabilities, they risk suffering a bank run that drains their USD reserves and puts them out of business.

> So I guess USD needs to be recognized by the US somehow?

Not exactly. It's possible to have a USD-denominated deposit account at a bank that h deposits at banks in the US. Most of the USD-denominated instruments in the world are not directly recognized by the US. But the international monetary system is hierarchical. Every USD instrument is an IOU for another USD instrument. If you follow the chain of IOUs, eventually, you'll get to the Fed.