the euro, in retrospect, appears to have been a misguided attempt to equalize the values for some very unequal assets, namely the bank deposits of strong countries and those of weak countries.
Modern governments seem to play this role in the monetary system. For example, deposit insurance serves to equalize the values of deposits in strong and weak banks.
What I see Cowen as saying is that the euro project inevitably entailed a sort of deposit insurance for banks. However, this was never formalized. No european-wide insurance fund was set up, and no risk-based insurance premiums were charged. Instead, it was more or less expected that each national government would prop up its own banks. But this assumption has proven unworkable.