I think that you need to formulate a more specific question. For fixed $k$, the $n$ is fairly irrelevant. Let $g(n)$ be a non-decreasing function which increases to infinity but exceedingly slowly (such as the inverse Ackerman function) then $f(n,k)=g(n)$ yields $k$ disjoint unequally colored edges with probability going to 1 (although for $k=6$, n will have to be unspeakable huge before $g(n)>5$).
At the other extreme, let $p_n$ be the probability that a random edge coloring of $K_{nn}$ (with n colors) yields a rainbow matching (with $n$ edges). I would have guessed that as $\lim_{n \rightarrow \infty}p_n=1$, and maybe that is true, but the small numbers point in the other direction. $p_1=1$, $p_2=\frac{7}{8}=87.5\%$ and $p_3=\frac{5090}{6561}=77.58\%$.
This still leaves a large middle ground with open questions (and even the $k=n$ case is not settled by what I wrote) later Indeed, it appears (see below) that right after $n=3$ it moves decisively towards $1$.
