Not only does the "13 button salute" refer to the 13 buttons on a Navy trousers, in US Navy slang, a "13 button salute" refers to the act of removing the trousers so fast that all 13 buttons come undone immediately, usually prior to intercourse. As such, the use of the term by the Union for a ship that has been taken over by hostiles and under duress is a somewhat derogatory term for that crew's predicament (i.e. they're "screwed").
Kaylon Primary assigns Isaac to read Alex Haley's novel "Roots", then tells Isaac to change his name, which is the first thing slave owners did to newly enslaved people. In Roots, main character Kunta Kinte is ordered to change his name to Toby. In the 1960s, when several African colonies broke away from European overlords, some native leaders ordered their subjects to stop using European names and adopt African ones.
Just as Admiral Halsey is named after the real life World War 2 Admiral William F. Halsey, the flagship of the Union Fleet is named after Admiral Raymond A. Spruance. The real Halsey and Spruance were close friends who alternated in command of the Central Pacific Fleet during World War 2. Between them, they either supervised or were in direct command of almost all the major naval battles in the Pacific.
Director and producer Jon Cassar said in an interview with The Orville Fan Podcast that he had to storyboard the giant end space battle after being told by Seth MacFarlane to "make it bigger than any Star Wars battle."
The Kaylons do not attempt colonization, but rather genocide: this closely mirrors the antithesis of the Borg's objectives, a group of races that have had cybernetic implants that assimilate other worlds, as established in Q Who (1989).