It depends on how the theft is charged. Generally, the statute of limitations on a misdemeanor is one year, and on a felony is three years. (Cal. Pen. Code § 801, 802(a).)
The individual cannot "press" charges - that is up to the prosecution. Yes, the "ordinary" statute of limitations mentioned are correct, but there can be tolling of the time depending on when they were discovered as well as longer statute of limitations for fraud situations.
The victim of identity theft should read and follow the steps in my legal guide on Avvo to protect themselves from harm to their credit and from debt collectors. Please see link to it, below. The Identity Theft Act is in Cal. Civil Code Section 1798.92 through 1798.97. It has a four year period from date of discovery to file a civil case against any "claimants." See Section 1798.96.
The Civil Act also refers to the Cal. Penal Code sections which explain what the District Attorney or other government prosecutors may do against the criminal who perpetrates the crime of identity theft, which may also be prosecuted for the crimes of fraud, criminal conspiracy, illegal wire tapping or tampering, mail fraud, RICO, theft, and any other number of crimes. There are also federal laws which may have been violated in the process of identity theft, for which the federal authorities may prosecute those crimes.