In a sports league, each team uses a set of at most $t$ signature colors. A set $S$ of teams is color-identifiable if one can assign each team in $S$ one of their signature colors, such that no team in $S$ is assigned any signature color of a different team in $S$.

For all positive integers $n$ and $t$, we aim to determine the maximum integer $g(n, t)$ such that in any sports league with exactly $n$ distinct colors present over all teams, one can always find a color-identifiable set of size at least $g(n, t)$.

The answer is $\lceil \frac{n}{t} \rceil$.

To show that the answer cannot be higher, consider the following construction: let team 1 have colors $1, 2, 3, \ldots, t$, team 2 have colors $t+1, t+2, \ldots, 2t$, and so on. Team $k$ will have colors $(k-1)t + 1, (k-1)t + 2, \ldots, \min(kt, n)$. This construction results in $\lceil \frac{n}{t} \rceil$ teams, so $g(n, t) \leq \lceil \frac{n}{t} \rceil$.

To prove that $g(n, t) \geq \lceil \frac{n}{t} \rceil$, construct a function $f(i)$ that maps each color $i$ to a team such that team $f(i)$ has $i$ as a signature color. This function must exist because each color is the signature color of some team. Let $S$ be the set of teams equal to $f(i)$ for some $i$. Construct a bipartite graph with teams in $S$ as left vertices and colors as right vertices.

There are clearly at least $n$ edges, one for each color. Since each left vertex (team) has degree at most $t$, there are at least $\lceil \frac{n}{t} \rceil$ teams in this set. Furthermore, because $f$ is surjective from colors to teams, Hall's Marriage Theorem applies, ensuring a perfect matching exists.

Thus, the maximum integer $g(n, t)$ is:
$$\boxed{\lceil \frac{n}{t} \rceil}$$