This is a "copywriter's usage". You know how in the US, for the super bowl (and also, say, The Simpsons seasons) .. roman numerals are used. It's always Super Bowl XXIII rather than Super Bowl 23. If you buy the DVD box set it's Simpsons IX rather than just Simpson 9.
You may ask -- why is that?
Quite simply, it sounds dramatic - a touch archaic
It's that simple. Using the roman numerals looks/sounds dramatic, archaic - it hints of a sort of scholarly importance, not unlike when an academic uses latin names, article references or the like.
So, quite simply, when people started talking about the browser war. (You can and do see "browser war" as well as "browser wars.") Some joker (a copywriter, or TV writer) was making a CNBC Original Series doco. on said commercial conflict. And, quite simply s/he decided that "Browser Wars" sounded more dramatic/funnier/whatever than just "the browser war."
(Indeed, that's precisely what Geo. Lucas did to tremendous effect in the title. It gives an archaic, dramatic, ye-olde-history-like feel to it -- it's so clever.)
{Imagine if the film had been called just "Galactic War." or "The Jedi War." Those just sounds like an ordinary action movie, about some action. No gravitas; no story-telling feel; no sweep-of-history feel.}
Indeed, there are any number of (cheesey) uses of the form in say advertising ("Winning the toilet-cleaner wars!!" sort of thing.)
Regarding the origin of the usage. It's a great question and I don't know.
Taking a wild guess, in Victorian times ("it sounds Victorian to me" -- I know you come to this site for hard evidence right? :) ) someone probably started referring to some messy ongoing set of wars with the plural. (As in "balkan wars".)
So I don't know the earliest usage; hopefully someone does.
But yeah, today in the USA it's one of those things "Like Roman Numerals" (or perhaps "using latin-esque naming"). You choose it because it lends a vaguely archaic sort of dramatic, story-telling-esque (or indeed, just humorous) air.
Star Wars is the ultimate (classy) example of this. As a cheesy usage, you can see it everywhere ("Basketball Wars! We take on East High tomorrow night!")
Regarding "is it a snowclone"? I urge you to set this question aside, because quite simply, the exact, absolutely precise, meaning of snowclone (or the earlier "catch structure", etc etc) is unknown. (For example, say someone was discussing what IS the meaning of snowclone: the sort of thing they'd discuss is whether the "_ _ _ Wars" format fits in to that meaning. So it's pointless us asking here, is it a snowclone.) If you want to assert that it's a snowclone - hell, you could assert it's the best and most primary example of a snowclone (given that Star Wars is a fundamental cultural artifact of the era) - if you wanted to.
Again the earliest use of the form is a great question - hopefully someone knows.