For the Monotremata example, Mammalia is a clade inside Synapsida, just as
Monotremata is a clade inside Mammalia. Thus, by extension, Monotremata is
a clade (one of many) inside Synapsida. The source of confusion for Anthony
Docimo, as far as I can tell, is that he is assuming that the word
"monotreme" is supplanted by "synapsid". This is conceptually incorrect.
As for the example of protists (which also surfaced on this list), the old
"Protista" is the poster child for just how BAD the Linnaean system was (and
still is). Not only did the "Protista" concept vastly understate the huge
phylogenetic diversity of single-celled eukaryotes, but we even had a
situation where certain protists were classified as BOTH animals and plants
(e.g., _Euglena_). The latter was a product of grouping organisms based on
"key features" or "gross physical similarities" (as Jerry puts it).
Euglenids moved and fed like animals, but they also possessed chlorophyll
like plants, therefore they got put into both Animalia and Plantae.