They are both correct designation and even intertwine between the luftwaffes manuals.
BFW was taken over by Willy Messerschmitt at 1938 which became Messerschmitt AG.

Niiiiiice. btw can you do the F6F Hellcat next please?

me 109 doesnt exists
bf stands for the company that produced the 109, but in 1943 germany had a plane designation revamp for new aircrafts which is the 2 letterq before the numbers is no more for the company but for the designer

thats why for post 1943 aircraft you get the me 262 for messershmitt and not bf
same for the ta 152 which is a development the fw 190 but named agter its designer kurt tank

but for older airvraft the old designation applied, which is why bf 109, fw 190 and such stayed

tldr me 109 is a false designation

I've always had a question about the 109.
Whats the difference between ME-109 And BF-109

wan oh nain hee free