It's called 'County Durham' so it's not muddled with Durham, the county town.

We do have a large amount of counties whose names are similar to the county town (Lincoln & Lincolnshire, Hereford & Herefordshire to name but two) so the addition of the word 'shire' could indicate county.

Mind you, I remember reading that a 'shire' has to be given that title - Devon has it but Dorset doesn't. The Dorsetshire Regiment was told to remove the 'shire' from its' name sometime after WWII but before its amalgumation with the Devonshire Regiment in 1959.

LTS