Was Voldemort's secret identity always known to the Ministry of Magic? Was it ever secret?

Question

Often called "He-Who-Must-Not-Be-Named" and "You know who", was it common knowledge at the Ministry of Magic that Voldemort was the same person as Tom Riddle?

Answer 1

I don't think that Voldemort was ever really a "secret" identity. Voldemort was a name that Tom Riddle was using as early as his time at Hogwarts, though at that time only amongst his closest "friends" (to the extent that he had them).

He pulled Harry’s wand from his pocket and began to trace it through the air, writing three shimmering words:

TOM MARVOLO RIDDLE

Then he waved the wand once, and the letters of his name rearranged themselves:

I AM LORD VOLDEMORT

“You see?” he whispered. “It was a name I was already using at Hogwarts, to my most intimate friends only, of course. You think I was going to use my filthy Muggle father’s name forever? I, in whose veins runs the blood of Salazar Slytherin himself, through my mother’s side? I, keep the name of a foul, common Muggle, who abandoned me even before I was born, just because he found out his wife was a witch? No, Harry — I fashioned myself a new name, a name I knew wizards everywhere would one day fear to speak, when I had become the greatest sorcerer in the world!”

_{Chamber of Secrets, Chapter 17 - The Heir of Slytherin}

When Voldemort returned to Hogwarts to ask Dumbledore for the position teaching Defense Against the Dark Arts, he was already using the name Voldemort exclusively and publicly:

“So, Tom… to what do I owe the pleasure?”

Voldemort did not answer at once, but merely sipped his wine.

“They do not call me ‘Tom’ anymore,” he said. “These days, I am known as —”

“I know what you are known as,” said Dumbledore, smiling, pleasantly. “But to me, I’m afraid, you will always be Tom Riddle. It is one of the irritating things about old teachers. I am afraid that they never quite forget their charges’ youthful beginnings.”

_{Half-Blood Prince, Chapter 20 - Lord Voldemort's Request}

I think, if anything, Tom Riddle would be considered the "secret identity", since that was the initial motivation behind the new name. Voldemort, as the Heir of Slytherin and with ambitions to be the most powerful and feared practitioner of the Dark Arts, would have wanted to ignore his Muggle ancestry entirely.

Answer 2

I'm not completely sure, but I believe that at the time of the events of Chamber of Secrets, it was not public knowledge that He Who Must Not Be Named was originally called Tom Riddle.

A relevant quote is in Chamber of Secrets in chapter 18. (Sorry for the long quote, but it does seem relevant.)

‘W-what's that?’ said Mr Weasley in a stunned voice. ‘You Know Who? En-enchant Ginny? But Ginny's not … Ginny hasn't been … has she?’

‘It was this diary,’ said Harry quickly, picking it up and showing it to Dumbledore. ‘Riddle wrote it when he was sixteen.’

Dumbledore […]

‘Brilliant,’ he said softly. ‘Of course, he was probably the most brilliant student Hogwarts has ever seen.’ He turned aruond to the Weasleys, who were looking utterly bewildered.

‘Very few people know that Lord Voldemort was once called Tom Riddle. I taught him myself, fifty years ago, at Hogwarts. He disappeared after leaving the school … travelled far and wide … sank so deeply into the Dark Arts, consorted with the very worst of our kind, underwent so many dangerous, magical transformations, that when he resurfaced as Lord Voldemort, he was barely recognisable. Hardly anyone connected Lord Voldemort with the clever, handsome boy who was once Head Boy here.’

Dumbledore effectively confirms this story in the Half-Blood Prince chapter 20.

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Further evidence is found in Philosopher's Stone chapter 17.

‘Call him Voldemort, Harry. Always use the proper name for things. […]

Dumbledore does not claim there that the proper name was Tom Riddle. So even though Dumbledore knew the Dark Lord's real name, he did not tell this to others, just like how he did not spread the other details he has found out about the Dark Lord's life.

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However, I believe that by the time of the events of the Half-Blood Prince, possibly even earlier, people would likely have learnt the Dark Lord's original name from Harry himself. In Goblet of Fire chapter 25 and 26, Harry has Rita Skeeter publish his biography, which would likely tell about this detail. It is possible that Harry mentions the events of the Chamber here.

Also, we know from Philosopher's stone that the Hogwarts students have found out about Harry's story very quickly, so it is very likely that they would also find out about the events in the Chamber of Secrets as well. This is especially true because in that year, every student was concerned with the attacks, so they would surely want to find out about the conclusion, whereas in Philosopher's Stone, most of the buildup part of the story was hidden from them.