They are virtual banks, they give you an account, but don't give you private keys, they control the coins.
They can impose spend limits, freeze your account, or even confiscate your balance.
They also ask for your ID, so you can't be anonymous, and all your transactions are tied to your name.
Use mobile or desktop wallet for transactions (e.g. Electrum, Mycelium), and paper wallets for storage.
If you don't want Big Brother to trace you, use a mix before sending to your wallet.
If you don't have sole and complete control over the private keys, you don't have bitcoin.
I want to make a comment about your post. Coinbase has the reputation of freezing accounts, blockchain.io is more like an HD wallet on the web. You can load your blockchain wallet using Electrum or MultibitHD by inputting your secret seed words as far as I know. The site itself does not control your coins. Please do not lump Coinbase and blockchain.io as the same type of service.