What the dickens is "vinger", and is it pronounced to rhyme with "ginger", or does it rhyme with "finger"?

It's the word for finger in Danish and Dutch. Sounds like venya. 

Could it be the word you heard is "Winger" :)

It started in early airplane days. fighter planes would often fly in a Wing formation With the ranking officer  in front and the 'lesser' pilots flying on either side and progressively further back (Geese usually form such wings too). The benefits of doing this (for pilots as well as geese) is that everybody other than the leader just need to keep an eye on the flyer in front and too one side of him, and maintain that position and that is a lot easier than navigating and reading every air-current himself.

The pilots on either side of the leader were called his "wing men" and those a bit further back were their wing men (just one for each) etc.
Even today we often call a squad of fighter planes a Wing.

But language changes and in popular use "wing man" became "winger" (a person assisting 'his leader')

You're not thinking of vinegar are you?

Well, without looking it up, I would say, neither. I would say it rhymes with dinger.  Just my call on it.