I've never understood the saying myself. I'm quite convinced that - faced with a photograph of my hand-back alongside similar pictures from another few males of my own age-group - I wouldn't have a cat in hell's chance of picking out my own! Different, I suppose, if you have a particular 'pattern' of markings, but in the absence of any such thing...? To me, it is one of these "obvious" sayings that isn't so obvious once you think about it.
The earliest recorded uses of the phrase date back only to the 1940s, so - although it sounds as if it might be quite ancient - it really isn't. Perhaps in olden days, more people were like me...ie it didn't even occur to them that they would recognise their own hand-backs!
I've also heard this as "I know it like the inside of my mouth" which I always considered to be quite disturbing as, in order to determine your own mouth from another's you'd have to use your tongue.
Obviously I'd have no objection if the mouth in question belonged to Keira Knightley or Natalie Portman for example, but what if it was Michael Barrymore???