A German state data protection agency has threatened Facebook's billionaire founder and chief executive Mark Zuckerberg with a €20,000 (£16,000) fine if Facebook does not allow Germans to have anonymous accounts on the social network.

In letters to Zuckerberg in California, and also to Dublin-based Facebook Ireland Ltd, the data protection commissioner for the northern German state of Schleswig-Holstein, Thilo Weichert, said the current rules violated German law by requiring users to provide their identities. "It is unacceptable that a US portal like Facebook violates German data protection law, unopposed and with no prospect of an end," said Weichert.

Under German law, media services, including Facebook, must offer users the choice of using a pseudonym.

It is not the first time that German privacy watchdogs have taken on Facebook. In 2011, Schleswig-Holstein banned local organisations and companies from using Facebook's "like" button, which it said allowed the site to monitor users; and the same year Hamburg's data protection authority ruled that Facebook's facial recognition feature violated German privacy laws. However, data protection experts said they thought it was unlikely that Facebook would comply with the latest attempt to force the site into line with German law.

"I think it is not very likely Facebook will change its business model for one country, or even just one region in Germany," said Jörg Hladjk, a lawyer specialising in data protection at Hunton & Williams in Brussels. "Just from a business perspective, this does not make a lot of sense."

Weichert said Facebook had filed for legal protection at the administrative court in Schleswig-Holstein, but there had been no change of position on Facebook's side.

A Facebook spokesman said the orders were without merit and a waste of German taxpayers' money. He added that the company would fight vigorously.