Notice how when listing out all three siblings, it’s perfectly fine to use triplets, yet when it comes to speaking of just one or two of them alone, this feels “somewhat intuitively wrong” to you.
The difficulty you’re having with triplets compared to twins is that it’s hard to mention someone’s twin in a way that leaves the other twin completely out of the picture, but with triplets it’s quite easy to do so, and this may feel odd. Your intuition fills in the missing member of a pair of twins, even if it is someone unnamed. It’s harder to do that with a set of triplets.
Nonetheless, the use you're not comfortable with can be found in published writing. It is rare, and most of it is recent works of fiction. Here are a few examples.
Cerea – Adventures in an Online World; Weby; 2007.
At four, Brian had travelled to Cerea, and last year he had gone to the mainland. His other triplet Gerhard, is the quiet type, saying maybe one word a day.
A Sorrow in Our Heart: The Life of Tecumseh;
Allan W. Eckert; 1993. [reference]
Finally, there was Lowawluwaysica, who resembled no one in the cabin, not even his triplet brother.
Children of the Unndis Sedna;
Archway Publishing, 2014.
With my wealth of insecurities, it helped me immensely to witness the innate confidence of my triplet.
Willian Campbell; 2014.
Considering my triplet's poor driving,perhaps a few other poor choices along the way.
Terry Spear's Wolf Bundle: The Heart of the Wolf, Destiny of the Wolf, and To Tempt the Wolf;
Terry Spear; 2010. [reference]
Finding out who killed my sister and why, and terminating him. I'm not interested in some alpha gray pack leader who just buried his mate who happened to be my triplet.
Touch Me in the Dark;
Patricia Rosemoor; 2011. [reference]
“Charlie, perhaps James has mellowed," Rose said of her other triplet who'd never left his native land.
Priests Under Pressure in Southern Moravia: History and Identity in Roman Catholic Polemics (1675–1615); Adam W. Darlange; dissertation thesis, 2009. [reference]
I also want to thank my parents Gary and Linda, Patty’s parents Bill and Marge, my older sister Amy, my triplet siblings Matthew and Add, and Patty’s siblings Kathy and Bill Jr.
It’s possible that the comparative rarity of triplet births is contributing to the unfamiliarity of the term. Twins number one in 30 natural births, but triplets are only one in 1,000 – roughly the square of the frequency for twins.
If you’re not comfortable with it, than don’t use it. Just know that some people do use it that way.