In the phrase "franco-prussian", what is the name of this construction using the contraction "something-o-something"?
There's no "something-o-something" happening, it's just "something-something" and one of the things happens to end in an "o".
"o" isn't a joining word, it's part of the word "Franco" which means "pertaining to France" (from Latin I believe). France may be the only country that gets this sort of word. With other countries you'd just use the standard adjectival form of the country name, eg
"English-Prussian relations", "Italian-Prussian relations" etc.