Your wallet stores digital coins and thus ultimately your computer holds your balance. The exchange keeps funds matching all unspent coins in a settlement account.
What if my wallet is lost?
Since the digital coins of value in your wallet are anonymized, the exchange can not assist you in recovering a lost or stolen wallet. Just like with a physical wallet for cash, you are responsible for keeping it safe.
The risk of losing a wallet can be mitigated by making backups or keeping the balance reasonably low.[1]
Ah. This is Chaum's DigiCash, from 1990.[2] A good idea, too early.
It's a system where coins are protected against double-spending by a database back-end. Which is how some centrally issued cryptocurrencies actually work.
The half-anonymity is interesting. The recipient is not anonymous, but the sender is. That may make it a politically acceptable substitute for cash.
It has the nice property that, unlike debit cards, etc., the recipient can't take more of your money than you give them.