The enigma of the 109's technical name is an enduring one.
But if we want to get super technical, it is a Bf109 by virtue of the RLM's assignment prior to Messerschmitt taking over Bayerisch Fleugzug Werke in 1938.
The Bf108, Bf109, Bf110, Bf161, Bf162, Bf163 and Bf165 were all BFW projects, not Messerschmitt AG.
The same happened with Blohm und Voss, who's company name was Hamburger Fleugzugbau GMBH and subsequently, their RLM designation was Ha before 1940.