There are three types of Augmented 6th chord the German, the Italian, and the French. Most chords are named for the intervals they contain or their function, but these seem like just arbitrary names for chords. Why are the augmented chords named this and is there an alternative name for each that better captures how they are constructed?
You would think the German, French, and Italian 6th chords are so named by the historical context in which they first appeared. (See also Neapolitan sixth chord, and the "Tristan Chord".)
However, the more I research this, it appears the names German, French, and Italian are likely arbitrary. Here are two citations that support this:
My recollections from undergraduate music theory (many decades ago) are that these were introduced in operas from each nation, but this appears wonky now.
However the "Neapolitan" 6th does have a specific origin:
I found a summary of a book that details "...Augmented Sixth from Monteverdi to Mahler" (Mark Ellis, Kirklees College and Huddersfield University, UK):
And just in case you want to see how the chords have been used please read what The Music Department at The University of Texas at Austin has here:
Note that the French sixth contains notes belonging to a whole-tone scale. Despite the strong temptation to draw a parallel to the French impressionist school, famous for their use of this scale (e.g. Debussy's Voiles), I concur with the previous poster that this is most likely a coincidence.