It's simple to answer because the OED does all the work for me. (I can't claim to understand American football and am completely clueless about rugby.) In the OED, scrimmage and scrummage are treated as variations on the same word, and the etymology is given as:If you are both a rugby and American football football fan, you will notice many obvious similarities between the two. I played rugby at school in England in the 70s and became familiar with the term scrummage, shortened to scrum in most usage nowadays. In the US, I have understood the word scrimmage to mean at least two things - 1. a term generic to many, if not all sports, meaning a practice game (a friendly in BrEng) 2. a specific American football term with which I'm not familiar.Can you comment on the root or roots of these? I feel confident that they share a common heritage, but I don't know for sure.
[Altered form of SCRIMISH n., the ending being associated with -AGE suffix. Cf. the parallel skirmage, obs. var. of SKIRMISH n.So, yes, they share a common origin. But the fun thing (for me, tireless defender of Englishes*) to notice is that we (again!) have a case of British people messing around with the language and Americans staying true to the original form--contrary to the popular stereotypes. Not that messing around with English is a bad thing, of course. After all, we wouldn't have poetry without some messing around.
This is now used primarily as a sporting term. The older i-form is common in all senses, and has become predominant in American Football, whilst the u-form is preferred in Rugby Football.]
* Actually, that's a lie. I'm a very tired defender of Englishes. The tiredness has little to do with the defending, though.