When I was young(er), I became obsessed with the First World War, also known as The Great War (trust me, it wasn't THAT great). In that context, I learned the meaning of the term 'batman' - essentially, it means 'servant'. One's 'batman' does domestic chores, cooks, cleans, etc. Comes from the 'public' schools (a.k.a. 'private' schools), boarding schools, Hogwart's and such, for all you illiterate and feckless youth. The 'batman' was an under-classman who performed duties for the upper-classmen.
So Alfred is Batman's 'batman'. And Batman is the 'batman' of JUSTICE.
If you thought Batman had anything to do with baseball, or Frank Miller, ferchrrissakes, then you're sadly mistaken.
I'm so glad I've met you in person and thus can hear this phrase in your voice, in my head. I can even see your nose wrinkling, and your finger pointing....
Get this - where I live in West London there is a hideous council estate at the rear of the BBC. It's called White City. Nestled away in a vile corner is a short row of dwelling units. Guess what it's called?
Aye...BATMAN CLOSE.
Hardly the sort of place you would find stately Wayne Manor.