Five years or fifteen years, it doesn't matter. Once you know someone or something a requisite amount of time, it's all equal.
But I will grant that Kirk and Mitchell were probably a closer relationship than my vague example. So, let me think... their are relatives I've known my whole life, friends from high school, co-workers from old jobs I still communicate with on the Facebook. Do I know, for sure, their middle names? I'm having trouble thinking of any middle names that I didn't actually contribute to making. I don't know my boss's middle name. I don't know her boss's middle name. I don't know my ex-girlfriend's middle name. I don't know any of my co-workers', sans one. And I don't think they know mine.
We take for granted that Kirk is known as "James T. Kirk" but maybe that's an affectation he picked up after this incident, perhaps because his best friend didn't even know it (in-continuity explanation!). Does Kirk know Gary's middle name? Does anybody know? It's not in any non-canon source that I can find.
Middle names and middle initials aren't that important in society anymore, and will likely remain so in the future. We went from Presidents like Franklin D. Roosevelt and Harry S. Truman and Lyndon B. Johnson to Jimmy Carter and Ronald Reagan and Bill Clinton. No one really knows about the E., W., or J. that used to be common knowledge.
I just think mistakes should be part of your calculations, and are far more likely, in-universe, than alternate continuities or multiple middle names that have never been mentioned before or since (or during). It's fun to speculate about the R., and you're not the first or last to offer an explanation. But the likely answer, in-universe, has always seemed glaringly obvious to most of us.
