Just shows how foresighed Leonardo de Pisa was when he described the Fibonacci sequence 800 years ago. The next/previous number is your km/mile equivalent, so
5 - 8 - 13 - 21 - 34 - 55 - 89 ...
I had noticed this, too. It's not exact, but near enough for approximation purposes. I've wondered for a long time if this is just a coincidence, or if there is a deeper connection between the two. Probably coincidence. Incidentally, there is a book called 'Nature's Numbers' by Ian Stewart which has a fascinating chapter on the Fibonacci sequence, relating it to the Golden Mean/Golden Section, the design of flower heads, snail shells and all sorts. Out of print now, but worth tracking down a copy if this kind of thing floats your boat.
Being brought up on Imperial units (ho ho), my mental arithmetic is quite good, so for miles/km I just multiply or divide by 1.6. Easy and near enough for most purposes.