Of course, I think both of these are Kessel Run explanations. I'm pretty sure Bond has a signature drink because he's the kind of bougie bro that likes to show off by giving bartenders instructions on how to make his signature drink. The actual cocktail never really mattered.
As I alluded to earlier in the thread, the emergence of the "signature" martini for Bond is a movie thing- the books have him drinking more scotch, bourbon, and.... everything else.
As for why it was popular, you have to remember that the UK was still in post-war rationing when the first Bond novels were released, and the idea of this type of consumption was probably some great escapism.
But looking more particularly at why the movie Bond drinks
vodka martinis... well, vodka was considered an exotic product in need of an image makeover in the post-WW2 environment. They were doing everything possible, including trying to hype new drinks like the Moscow Mule. And trying to get business people to drink vodka because it didn't have the tell-tale odor of other drinks (so they claimed).
But you know what really worked? Watch Dr. No- that's right, not only the origin of the phrase "shaken, not stirred," but it has a very clear and shot of Smirnoff's vodka.
Marketing does amazing things, don't ya think? Now, don't even get me started on the people drinking Grey Goose.