Regardless of whether you use Coinbase or your own wallet, Bitcoin and most other cryptocoins are not "anonymous" as it would be traditionally considered. See "A Fistful of Bitcoins Characterizing Payments Among Men with No Names" [1].
The original bitcoin whitepaper doesn't mention anonymity or privacy anywhere. What it does show is a partition between identities and transactions. Transactions can take place without ever naming an identity. But we've learned in the meantime that this partition is not sufficient, in practice. All it takes is one transaction where some part of your identity or personal information is disclosed (shipping address, e.g.). From there, the transaction history could be followed to determine where you got the money from (strip club, silk road, etc). Having multiple independent addresses mitigates some of the impact but it's not enough to describe it as "anonymous".
Case in point: "Stealing bitcoins with badges: How Silk Road’s dirty cops got caught" [2]. TL;DR Chainalysis followed the transactions on the ledger.
