That's a common misconception, they are actually named after the famous jazz instrument the saxophone.

Lots of things are possible, fewer are likely, even fewer attested, even fewer attested without doubt.

Though it'd have a special irony to it: Some Roman asking a bunch of probably Austro-Bavarian or Alemanic warriors "who are you" and getting the answer "Germannen" -- "spear men". Were Italy an Island they would've said "sailors".

Probably more likely than Romans observing those tribes, noting more or less bog-standard javelins, not exactly a rare or particularly notable technology, and saying "hey let's ask them what they call javelins and add a -man after that for no apparent reason". Even if they specialised their tactics etc. around those spears would you call Greeks "Phalanx people"?

According to German Wikipedia, this theory, Germans are men with a Ger, is longer up to date.