Is this reduction from 3D-MATCHING to PATH SELECTION invalid?
The "additional constraint" is part of the problem definition:


  for a given directed graph and a set of paths we ask if it is possible to select at least k paths so that no pair of paths share any node.


The key is "and a set of paths". The Path-Selection Problem has a set of paths from which you are allowed to chose as part of its input.

Note that the definition is kind of strange, since the fact that a graph is involved seems irrelevant (we could always take the graph to be a clique, which doesn't change the answer). However, if the graph has special structure, then the problem can become tractable (e.g., if the graph is a tree the problem is polynomially solvable).