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Source: ECB
It works by having a central fund to back the money that qualifies for the deposit guarantee, however said funds only contains 0,8% of covered deposits. Although this might seem small, this is still a large amount of capital (~40 billion euro), and should be able to cover all deposits during a major financial crisis (like 2008) according to this research (ECB funded).
Similar with the US FDIC:
The FDIC is primarily funded through assessments, which are insurance premiums paid by FDIC-insured institutions. These assessments are based on the balance of insured deposits and the risk posed by each bank. Additionally, the FDIC’s Deposit Insurance Fund is invested in U.S. Treasury securities, earning interest that supplements the premiums paid by banks.