The names Americans used for Japanese planes weren't established until 1942. From the outbreak of war until then various systems were tried out and none were altogether satisfactory. In mid-1942 a standard was set up by USAAF Captain Frank McCoy. Initially fighters were given male names of one or two syllables and bombers were assigned similar female names. Later, female names beginning with the letter "t" were applied to transports, and tree names were given to trainer aircraft. This system simplified the previous systems and caught on quickly. Wikipedia has a decent list. So there was Zeke for the A6M Zero and Nate for the Japanese Army Type 97, for example.

I'm trying to think of other examples, but the only one that comes to mind is the American name for the Japanese Ohka. This was a rocket-powered kamikaze aircraft that was basically a guided missile with a human as the guidance system, made to be launched from the belly of a bomber aircraft. Instead of calling it the Ohka or another name, the Americans called it the "Baka"--Japanese for "fool" or "idiot." In the other histories of WWII aviation I don't recall other nicknames given to enemy fighters, but perhaps someone else will recall something that I missed or have forgotten.

The Germans called the P-38 Lightning "the fork-tailed devil."

Supposedly the Japanese called the Corsair "The Whistling Death"... there's some question about that.