Currency denominations seem to be efficient* when they follow roughly half-an-order-of-manitude increases -- e.g. $0.05 to $0.25 to $1 to $5 to $20 to $100.
Adding mere doublings in there just complicates things for no good reason. There's already no particularly good functional reason for dimes or $10 bills, except we seem to want a denomination for every power of 10 -- it's more of an aesthetic thing.
But there's absolutely no practical reason for a $2. There's no functional reason and no aesthetic reason either.
* Finding a practical balance between carrying a minimal number of bills for any given total value (where powers of 2 would win), versus minimizing infrastructure like sizes of cash registers (where powers of 10 would save space in layout).
(Edit: maybe the case is different for heavy coins though? E.g. $10 in $2 coins will weigh significantly less than in $1 coins in your pocket. On the other hand, the US already mainly uses bills for $1, which is why $2 makes no sense in the US.)