I like 'Navajo: Béésh Bich’ahii Bikéyah ("Metal Cap-wearer Land"), in reference to Stahlhelm-wearing German soldiers.'

This apparently confused the English so much that they called the inhabitants of the Netherlands "Dutch".

It's especially common when the two sides have difficulty communicating: "What do you call yourself?" "Ger man." (Spear man [1]) "... henceforth the lands east of the Rhine shall be known as Germania."

[1] This is only one possible etymology.

My guess for Germany, which certainly seems the weirdest, is that it's a result of that process happening and the name getting fixed at different times for different languages, combined with the region having a rather complicated political history.