Red, especially deep red, tends to be the most prominent color on multicolor maps, and comes to the visual foreground. So mapmakers use it to symbolize whatever they want to emphasize most. I can't read the map labels here, but it probably was intended to show how many of the world's nations were part of the Axis. The association of red with both Nazi and Japanese flags and insignias was more a coincidence; after all, British, Soviet, and American flags also use red.

Red as a "bad" color is very much a recent and tertiary association, from 20th century references to communism as "red." Red was usually the first color used when printers, particularly newspapers, magazines, and map publishers moved beyond black-and-white. It really adds emphasis, and gives you the most bang for your second-ink buck. The very earliest road maps, for turn-of-the-century bicyclists, used red to highlight the good cycling routes. Maps for motorists followed suit.

As an aside: color hierarchy can be a tricky matter for today's cycle-routes cartographer. Departments of transportation, who pay for most cycling maps, love the idea of red-means-bad; yellow-means-caution; green-means-good. They think it's mnemonic. But on a typical cycling map, that means you're applying red to the high-speed busy highways, and you end up distributing a map that emphasizes all the places you shouldn't bicycle. As a cartographer, I've had many intense conversations on this subject with clients through the years.