If you replace a boat piece by piece, is it the same boat?

The water does not know the difference, that's all that matters. 

If you replace the broken handle of your grandfather's axe, is it still grandfather's axe. Or replace the business end?

"Trigger's Broom" has become a fairly common saying in Britain for that principle!

("Trigger" is the road-sweeper character's nick-name in the BBC TV sit-com Only Fools and Horses, and in one episode we learnt he'd replaced both the head and handle, separately and many times, "but it is still the same broom".)

It is. Unless, of course, it isn’t.