Here is a possible title **matchings in hypergraphs** This is just a restatement but it might help with finding  references. For $T=2$ one has the usual graphs and there are good algorithms for finding maximum size matchings. For larger $T$ finding matchings is  (in general) NP-complete even for $T=3$. It may be that there are special features of your problem that bring things into reach.

For $T=2$ make a simple undirected graph with vertices $1,\cdots,N$ and edges all pairs $(i,j)$ which are allowed ($i \in S_1 \text{ and } j \in S_2$ or vice versa). A matching is a set of vertex disjoint edges (also called an independent edge set). If $KT=N$ then you are asking for a perfect matching.

If $T>2$ one can consider a hypergraph whose hyper-edges are all legal $T$-tuples and look for a matching.

Of course it is more compact to list your sets $S_i$ than to list every possible $T$-tuple. Your example above is the complete graph $K_6$ missing only the edges $(1,5)$ and $(3,6).$ The full graph $K_6$ has $15$ perfect matchings, $3$ using each edge. Of these, $10$ work for your problem.

Here is a slightly more promising idea: Make a bipartite graph with $N+T$ vertices labelled $1,\cdots,N$ on one side and $S_1,\cdots,S_T$ on the other. Draw an edge for $j$ to $S_t$ when $j \in S_t$. Then a maximum size matching of this graph would have at best $T$ edges. So you could look for $k$ disjoint $T$-matchings. This has the added structure that each tuple is essentially ordered.