A bitcoin isn't a "thing". It's an entry in a public ledger that says "wallet X paid wallet Y 0.001BTC". All* the wallets start out at 0 and if you want to see how much money wallet Y has now, you start at the beginning and add all the payments into it and subtract all the payments out of it. Example:
has 0 BTC balance after receiving 3 payments and making 3 payments (the tiny fraction of a BTC missing each transaction is the fee paid to miners to process it).
As for anonymity, I normally have no way of telling you who 1ENYmn1eCWPa4MFD4VU9wUFqLrzPcqUgaY is. But if one of those payments there was made from an ATM that converts money to BTC (and takes a photo) or one of those debits was sent to a drug dealer who mailed the drugs to the wallet owner's home address, then got busted by the cops who got a customer list, then it could be figured out.