Name of ML discipline for trainable clusterer - supervised clustering, or group ID assignment
Perhaps you are looking for some combination of Meta clustering and Ensemble clustering?

Meta clustering


  A sampling-based approach that searches for distance
  metrics that yield the clusterings most useful to the user


...


  
  Generate many good, yet qualitatively different, baselevel
  clusterings of the same data.
  Measure the similarity between the base-level clusterings
  generated in the first step so that similar clusterings
  can be grouped together.
  Organize the base-level clusterings at a meta level (either
  by clustering or by low-dimension projection) and
  present them to users.
  


Ensemble Clustering


  In statistics and machine learning, ensemble methods use multiple learning algorithms to obtain better predictive performance than could be obtained from any of the constituent learning algorithms alone. Unlike a statistical ensemble in statistical mechanics, which is usually infinite, a machine learning ensemble consists of only a concrete finite set of alternative models, but typically allows for much more flexible structure to exist among those alternatives.