Newfoundland at the time of the story was a self-governing British colony, hence the "Republic of..." title. That was also why the detective's badge was confiscated; the Canadian constabulary had no jurisdiction there.
The title is also reference to "The Republic of Doyle", another popular CBC show set in Newfoundland. Even the main actor Allan Hawco guest stars as "Jacob Doyle", a play on his character of the same name Jake Doyle.
The character of Jacob Doyle is, more than once, referred to as a "sleiveen" or "sleeveen." It is a word used in Ireland and Newfoundland referring to a sly, smooth-tongued person, a rogue or a trickster, usually from a rural area.
Throughout the series Murdoch meets many famous personalities of the day, whose presence in Canada is a poetic license. However Guglielmo Marconi was actually in Newfoundland in 1901, when he had the receiving station for the first transatlantic wireless telegraph constructed on top of Signal Hill.
Detective Murdoch inadvertently invents the rudiments of the game Scrabble while trying to determine the possible letter combinations contained in the bottom third of a torn piece of paper.