Yes, Voldemort is the only person who possesses others.
There are three instances of possession in the books. All of them involve Voldemort and all of them were made possible, on some level, because of the Horcruxes. Nobody else possesses anyone else, or is said to have the ability to do so. In Philosopher's Stone Voldemort possesses Quirrell. In Chamber of Secrets he possesses Ginny. In Order of the Phoenix he possesses Harry. Since the questioner has signalled that they're interested in the third instance I'll focus on Voldemort's possession of Harry in the Ministry Atrium.
As for how Voldemort was able to possess Harry, it was almost certainly down to the close connection between the pair.
"“What you must understand, Harry, is that you and Lord Voldemort have journeyed together into realms of magic hitherto unknown and untested...Having ensured this two-fold connection, having wrapped your destinies together more securely than ever two wizards were joined in history, Voldemort proceeded to attack you with a wand that shared a core with yours."
(Deathly Hallows, Chapter 35, King's Cross).
Voldemort had Harry's blood in him. Harry had a portion of Voldemort's soul in him. I think the strength of their connection was such that Voldemort could possess Harry, and that he couldn't have possessed anyone else in a similar way. The soul portion was critical, I think; Harry may not technically be a Horcrux but yes, close enough, the possession happened because of the Horcruxes. If Voldemort hadn't fractured his soul to the extent that another person had a part of it inside them then I doubt that this kind of possession would've been possible.
It's worth noting that, as far as Harry was concerned, this was a one-way process that couldn't be repeated in reverse.
"He did not know that you would have ‘power the Dark Lord knows not’-”
“But I don’t!” said Harry in a strangled voice. “I haven’t any powers he hasn’t got, I couldn’t fight the way he did tonight, I can’t possess people or - or kill them-”
(Order of the Phoenix, Chapter 37, The Lost Prophecy).
To briefly deal with the other two possessions, they were different. The possession of Ginny was caused by her close dependence on the diary-Horcrux. The soul-portion in the diary developed that dependence until it could possess and control her. The possession of Quirrell was (supposedly) voluntary; Quirrell was meant to be OK with Voldemort possessing him. It was only possible because Voldemort didn't have a body at the time and was somewhat spectral.
"I was ripped from my body, I was less than spirit, less than the meanest ghost...I was as powerless as the weakest creature alive, and without the means to help myself."
(Goblet of Fire, Chapter 33, The Death Eaters).