Given a set $S$, is there a name for the following:
$$f(S,N) = \bigcup\limits_{k=0}^{N}S^{k}$$
For example for $S=\{x, y, z\}$ and $N=2$ the result would be: $$\{(), (x), (y), (z), (x, x), (x,y), (x, z), (y, x), (y, y), (y, z), (z, x), (z,y), (z,z)\}$$
Is there any way to relate it to a notion/terminology in mathematics?
Yes, it is standard to denote this set by $S^{\leq N}$.
It is always best to explain the notation you use in words (as Punga suggests in the comments) the first time you use it, especially if it is less familiar.
The set of all finite sequences from $S$, with no bound on their length, is written $S^{<\omega}$ or $S^*$. The first notation is used (among others) by set theorists - here $\omega$ is the least infinite ordinal. The second notation is used (among others) by computer scientists - here $^*$ is the Kleene star.