your identity is the private account key. which you cannot transfer, and a name is just that. a name. I would never sell a name I'm actively using for trades, however, I like the freedom of choice.
I would say your identity in this system is your account ID. The following are my thoughts regarding identity, accounts, names, and keys.
Your identity should definitely not be your private key, as you should be able to update those for security reasons. Even the owner key held in cold storage may need be to changed from time to time. For example, if the owner private key was derived from both a brain key stored on paper in your home AND a memorable passphrase only you knew in your head, then even if the the brain key was compromised (say you suspected the paper backup was accessed during a home invasion) you would have a very reasonable amount of time to change the key before the thief could brute force the passphrase even if they had supercomputers at their disposal (assuming the passphrase wasn't ridiculously simple and was also not used anywhere else). The more paranoid could also just normally assume the brain key was compromised and regularly change the owner private key frequently enough to avoid having even a moderately large cluster of modern computers able to likely brute force the passphrase in time. The user would either use a backed up brain key and the passphrase to derive the existing private owner key which could be used to sign a transaction offline or would use their fallback permission system (a quorum of pre-selected friends and family signing a proposed transaction) to officially change the account's owner key to something else (derived using a new random brain key and newly chosen unique passphrase). Your identity (account ID) would still be preserved through this process.
It also problematic if your name is your identity. What about someone who changes their name and wants to change their account name correspondingly to reflect their new name. Clients could notice this change on the blockchain and know that an account they had in their address book was formerly known as the old name but is now known as the new name. The blockchain protocol could require some set period of time (like 2 weeks) before an account name that was transferred or revoked could be activated or reused by another account in order to reduce the likelihood of the edge case where a user who recently heard about someone's account name (but has not yet added the user to their address book) sends unsolicited funds to that account name but it ends up going to the new user rather than the intended old user. This problem can become even less of an issue if some of the least significant digits of the account ID are provided along with the name to be used as a check to make sure funds are really being sent to the intended recipient.
Finally, as I mentioned before, I think it should be possible for a user to have an account without a name. In that case, the user still has an identity (it's part of the account). They just don't associate that identity to a public human readable name (perhaps for privacy reasons). Setting a random account name as a substitute for this seems pointless and only seems to bloat the blockchain and prevent the separation of account registration fees from name registration fees (meaning without the separation we wouldn't be able to have moderate fees for non-premium names, to prevent spam and some squatting, while still letting users create accounts for very cheap fees). Sharing the unique account ID is a perfectly fine way to share your contact information with someone so that they can add you to the address book (it is less cumbersome than addresses), especially if the clients generate and verify a dictionary word checksum derived from the account ID that should be shared along with the ID (or even reversibly convert the account ID + checksum into a sequence of dictionary words which are much easier for humans to read and remember than a long number). Of course, when computers directly share the account ID (via web links or communicated via NFC or Bluetooth between mobile devices) users would not need to even be concerned with the underlying details of the account ID. In such cases, where a recipient is referred to in the user's GUI client using an ID (in some form) of an account without a name, the recipient would need to first be added into the user's address book and be given a local name (and other local information like a picture) that helps that user identify that recipient in the future. Also, even for an account with a name that is added into the address book, the GUI client should still allow the user to specify a preferred local name (and other local information like a picture) for convenience. There should be clear separation in all parts of the GUI that use a name to refer to an account between a local name (which could have full Unicode support) and account's global name registered on the blockchain.