All it does is prove that someone controls the private key to that particular bitcoin address.
I've seen this a couple times. It does
not prove that someone owns a particular public key: it proves that the owner of the particular public key approves of the message that was signed. It's similar to [the intention of] a regular hand-written signature -- you don't sign blank sheets of paper to prove who you are, but you do sign sheets of paper that identify something you agree with.
I think it really depends on the content of the message.
If the message just contains hex-like lists of addresses or the like without any meaningful statement, then the signer just prooves that he controls the private key to the address.
If the message contains a meaningful statement like e.g. a contract, the signature prooves that the owner of the public key approves of the message.
I want to point out, though, that it will still remain difficult to proove e.g. in court that the claimed owner of the public key is really the 'real' owner. If e.g. Jim's private key was stolen, the thief could sign messages instead of Jim. So Jim could claim in court that his private key was stolen and that somebody else (not Jim) signed the message with the stolen key. Regardless if a court or people will believe it, I just want to point out that prooving the ownership of the private keys is not the same as prooving your identity.
In some European countries the governments have started to incorporate the ability of electronic signatures into the identity cards. This is intended to identify yourself online in combination with a PIN. A thief would need to steal the ID card and get hold of the PIN. In the above example of Jim, as soon as Jim detects the theft, he will inform the ID card issuer to invalidate the electronic signature. Later in court he will be able to proove that he informed the issuer, freeing him of any responsibility of his stolen electronic signature. Failing to do so will make him liable in court, because he had the ability and the responsibility to inform the card issuer.
Since invalidating public keys is not possible with bitcoin, Jim will have a chance to claim in court that somebody else signed the message.