Elf apparition is unique to elves and is not the same as a wizard's. Hence it may not be detectable by the Ministry, the same way Wizard Enchantments cannot prevent them from Apparating or Disapparating.
House-elf apparation is unique to house-elves; it is a different form of magic:
“Elf magic isn’t like wizard’s magic, is it?” said Ron. “I mean,
they can Apparate and Disapparate in and out of Hogwarts when
There was silence as Harry digested this. How could Voldemort
have made such a mistake? But even as he thought this, Hermione
spoke, and her voice was icy.
“Of course, Voldemort would have considered the ways of house-elves far beneath his notice, just like all the purebloods who treat
them like animals. . . . It would never have occurred to him that
they might have magic that he didn’t.”
Ron says that House-Elf's magic is unique to elves, at least in regards to apparation. Although Ron may not know much about house-elves, Hermione confirms that "Elves have magic that wizards do not."
Another proof that the magic is different: Whenever house-elves apparate in the books, they never need to spin on their heel.
What is the source of this unique magic? it seems inherent to the magical creature:
“The house-elf’s highest law is his Master’s bidding,” intoned
Kreacher. “Kreacher was told to come home, so Kreacher came
home. . . .”
Ron seems to have understood as well that by House-elves it's instinctual:
“Wish I could Disapparate like a house-elf,” said Ron, staring at the spot where Dobby had
vanished. “I’d have that Apparition Test in the bag.”
(Of course, this could be interpreted differently.)
It seems that Wizards cannot stop a house-elf from Apparating or Disapparating. Even the Malfoys do not have a charm stopping house-elves from apparating into their Manor, and presumably, unlike Voldemort, they would've considered the possibility, since they themselves owned Dobby in the past.
Dumbledore is in charge of the Hogwarts Apparation charm:
“As you may know, it is usually impossible to Apparate or Disapparate within Hogwarts. The
Headmaster has lifted this enchantment, purely within the Great Hall, for one hour, so as to enable
you to practice.
Yet he too does not enchant Hogwarts to disallow elf apparation.
From all these cases, and from the fact the Ministry did not detect Dobby's apparation, it seems apparent to me that Wizards are in fact unable to prevent or detect House-Elf Apparation.
As for the Hover Charm, perhaps it was a wizard charm that Dobby learned from the Malfoys, and thus detectable.
As an aside: Interestingly, Elves aren't the only animals who can apparate without human restriction. Phoenixes can too:
Fawkes circled the office and swooped low over him. Dumbledore
released Harry, raised his hand, and grasped the phoenix’s long golden
tail. There was a flash of fire and the pair of them had gone.
“Where is he?” yelled Fudge, pushing himself up from the ground.
“Where is he?”
“I don’t know!” shouted Kingsley, also leaping to his feet.
“Well, he can’t have Disapparated!” cried Umbridge. “You can’t inside this school —”