First you have to realize that an “identity” is nothing more than a construction, whether it be social, legal, or otherwise. Therefore, in most cases, it is very easy to change and alter it. In most cases your identity only exists in an active negotiation with another entity. (It becomes harder to change your identity when it is nothing more than an entry in an electronic database somewhere. Nearly all of us here have an identity like this; an identity which is constructed based on a number of digital interactions, none of which you are aware are being stored or even being made in a demographic. Very interesting field of research, actually.)
For the OP, I guess you are just trying to leave behind the identity you’ve constructed from your birth until today. I do not see why that would be illegal. Where you encounter a problem is in trying to legally create your new identity. Creating a new social identity would be easy enough: move to a new place, dress, act, and talk according to how you wish your new identity to be perceived and presto, you have a new social identity. However, if you wish to create a new legally recognized identity, you’ll probably have to commit fraud at some level (either deliberately or just through simply not offering certain information).
Regardless, many people have changed their legal and social identities over time. See chapter six in Sorting Things Out for a discussion of how individuals managed to “pass” as a different racial category during Apartheid. The most obvious method was: “first obtain employment in a whites-only occupation that is not too fussy about identity cards… move to a mixed neighborhood, and quietly join the local white associations… Over time, this establishes a track record that can be used as leverage for reclassification [to a new identity] based on general acceptance and repute” (Bowker and Star, 1999, p. 216).
The same sort of method would need to be used in a Western country. You would have to build up personal relationships that believed your “presented” identity is actually your true identity. Succeed in doing that and you have a newly constructed “social” identity. Now, with a firmly established social identity you would slowly proceed to constitute a new “legal” identity. Could you do it without ever committing fraud, on at least at some level, probably not. Could you do it? Definitely, people are actively doing it everyday. Is it illegal? Depends on what you consider your identity to be. Is your identity that which others, whether social or institutional, assign to you? Or is your identity you, and only you?