A pair is a set of two things of the same type that are used together. What do you call a set of three such things? What about four, five, etc.?
Three: Triplet, trio, triad, or three-of-a-kind
Four, five, six: Quartet, quintet, sextet. Note that these words are uncommon and sound quirky or formal.
It's safe to call 'a set of three' or whatever number you want. Why? Because if the context is not clear, it may sound odd.
Here is the list of, for example, set of 'three' in various cases. Note that for each condition, the word is different. While for three kids delivered it's triplet, for a card game, it would be 'trio.' Furthermore, in cricket, it'd be 'hatrick' when three batsmen are out; 'trilogy' for a tri-series movie.
So, again, to be on a safer side, call 'set of ' when the context is not clear. Say, you buy a set of three shelves from some store.
Threesomes. They say pairings in golf (sometimes twosomes) when there are two in a group. They need to come up with a similar word when there's three. My suggestion is triplings.
Trilogy is what you call a work with a set of three
Triumvirate: any group or set of three.