Expressions in Tim Minchin's "Storm"

"I confess a pigeonhole starts to form": This is referring to the third definition of pigeonhole here, "a neat category which usually fails to reflect actual complexities". He is saying that he is beginning, in his mind, to ascribe to her the characteristics of a particular stereotype (the details of which become apparent later on). The term comes from the use in offices of shelves with compartments for different types of form - so there will be a pigeonhole for form 43A, and another for form 66C; the term is also used in UK offices for what I believe in the US would be called a mailbox. Of course this in turn comes from its physical resemblance to the original meaning, "a hole for pigeons to live in".

"diplomacy dike": He is using the word "dike" (spelt "dyke" in the UK - that is not an error in the subtitling!) in sense 2b here: a barrier or dam to hold something undesirable back. By "diplomacy dike" he means his diplomatic self-restraint, which generally serves to keep his tongue from voicing his insulting thoughts, and in this case is holding back a barrage of abuse (that comes out later in the poem).