The Manchester United star sought protection against claims of an alleged extra-marital affair with Welsh model Imogen Thomas. At a hearing at the high court in London, Giggs consented to the lifting of the anonymity injunction.
The father-of-two was widely revealed on social network websites online, while Liberal Democrat MP John Hemming used parliamentary privilege to name him.
His lawyer Hugh Tomlinson QC said that Giggs, 38, is seeking damages for the re-publication of The Sun's original information, which he claimed had "generated a large media storm".
"The claimant's name is in the public domain contrary to court orders," Tomlinson told the court. "The claimant has consented to the removal of the anonymity order completely.
"He has suffered damage and distress by the chain of events that has been set off by the publication of the article in The Sun. We say the printing of information on the front page of a national newspaper can give rise to an action for misuse of private information."
However, while the anonymity order was originally lifted on February 1, it is believed that an "administrative error" by solicitors meant that the newspaper was not informed of the decision.
He added that the damages claim was "providing effective protection" for his client's right to privacy under the European Convention on Human Rights.
Mr Justice Tugendhat reserved his decision on whether the case should go to trial, revealing that the action had already technically been struck out, after solicitors for Giggs failed to keep the action alive. Tomlinson described Schillings's mistake as a "silly and unfortunate error".
Big Brother star Thomas celebrated in December when Giggs dropped accusations that she had attempted to blackmail him.
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