Is there a specific search paradigm for finding pairs in a set?

Question

I'm dealing with a very common problem in computer programming that involves, for example 4 people to be divided into 2 pairs. Mathematically, this is just a permutations problem, and the number of possibilities would be 4!/(2! * 2^2) = 3.

In computer programming, when searching for these specific solutions, a programmer would just recurse through the possible solutions. For example --

int solve(void) {
   int i, count = 0;
   for (i = 0; i < n; i++) {
     if (partners[i] == -1) {
       break;
   }
 }

 // if all the pairs are matched with each other
 if (i == n) {
   return check() ? 1 : 0;
 }

 for (int j = i + 1; j < n; j++) {
   if (partners[j] == -1) {
       partners[i] = j;
       partners[j] = i;

       // recurse back through to find all possible sets of pairs with this configuration
       count += solve();
       partners[i] = partners[j] = -1;
    }
 }
  return count;
}

Now, what I found interesting was that when you draw a permutation tree diagram, you see that the unique solutions for this problem arise naturally in this pattern: Permutation Tree. This pattern goes on as the number of people increases. Taking all that into account, my question is, is this simple recursive function a unique search paradigm (like BFS or DFS) that models the behavior/pattern seen in the tree? If not, could an algorithm be designed that models that pattern?

I'm a high school student, and I'm comfortable with basic and intermediate algorithm design concepts, but I'm still learning, so if you could, please explain your answer in more depth than usual.