When morphological traits (i.e., physical traits) are used to define the groups, the method is usually referred to as phenetics or, more broadly, cluster analysis.  When the traits are genetic, one clustering method employed is referred to as neighbor-joining.  The tree so produced is usually called a dendrogram or phenogram.  There are many methods of clustering in use.

In genetic genealogy, a clade usually refers to a group united by possessing the same SNP mutations.  By logically deducing the chronological appearance of SNP mutations, it's possible to produce a hierarchical branching diagram called a haplotree, which is, by nature of its construction, a cladogram.  The tree itself and each of its individual branches is a clade, and the method used to form such clades and trees is called cladistic analysis or simply cladistics.  At each branch in the tree a determination must be made as to the "polarity" of the trait involved, that is, which state is ancestral (-) and which is derived or descendant (+), so their chronological order can be logically deduced.  Cladograms may be node-based (at the forks of the tree) or branch-based (on the limbs of the tree), and may or may not include time as a factor, usually indicated by the lengths of the branches in the diagram.

When genetic data are used, the cluster often has a high statistical probability of also being a clade (of having a single common ancestor), but there always remains the possibility that it isn't a clade due to convergence or parallelism (i.e., coincidence). 

Proving a group is monophyletic is important because the single greatest problem in evolutionary biology is detecting convergence and parallelism, which will cause organisms or haplotypes to resemble one another, despite having different ancestral origins.  Convergence and parallelism are less of a problem with genetic data than they are with traditional morphological traits, but the problem has not disappeared.

There needs to be something "outside" the cluster that independently insures its monophyly.  In the case of genetic genealogy, pinning STR clusters to a SNP cladogram (i.e., using STR testing and SNP testing in combination) is one way of assuring the clusters are also clades.  Another way is using paper pedigrees to determine the polarity of STR mutations, though it's only useful in genealogical time.  Such proofs, where independent forms of evidence support the same conclusion, are often termed, "elegant," because of their scientific rigor. 

Lastly, and especially, I would like to emphasize that SNP status is not a mere "attribute" of the STR cluster.  The organization of SNP mutations into a cladogram (haplotree) supercedes the organization of STR mutations into clusters.  You are a SNP clade, first, and an STR cluster, second.