No personally identifiable information is stored on the Bitcoin blockchain. It is true that there are some speculative methods that may identify the originating IP of a Bitcoin transaction but, those use an educated guess method and are unreliable - particularly if a node connects to the Bitcoin network using Tor then the educated guess method is unusable in determining the originating IP.
In a broad sense, the issue of privacy is universal. If you purchase goods or services online you must ordinarily provide a name and often a delivery address. If you purchase in store this may not be necessary. If the merchant shares the information that they have about a transaction (or is forced to hand it over to a government) then with payment details the details can be mapped to a set of utxo's on the blockchain which funded your purchase transaction, and any change address. This is terrible for privacy.
As I understand it this situation is improved with Monero by the use of stealth addresses, nonetheless, the details that are retained can still be handed over.
For myself, I see the privacy concerns of Bitcoin as a non-issue - just use Tor. You cannot avoid informing a merchant of information necessary to complete a purchase. Pieter Wuille has a more academic view and states:
Tor does not hide account linkage or amounts. Without very proactive measures anyone you interact with can probably make a reasonable guess how much money is in your wallet (by distinguishing change from payments), what software you are using, and with further analyses what sites you interact with. All of these are terrible properties for a currency, threatening its fungibility. Note that privacy is always optional - even when using stealth addresses or confidential transactions, you can prove to anyone what your balance or transactions are. It just becomes opt-in rather than opt-out. – Pieter Wuille
Whilst I can appreciate how such linkages are mapped, it is beyond my comprehension how such linkages are useful if Tor is utilised since without an originating IP it seems speculative to link further transactions over those immediately attached to a particular transaction into the available information set, I do not think it is useful, nonetheless, I will take Pieter at his word.
This answer has been edited in view of the discussion by Pieter Wuille in the comments.