A haj színe - the color of the hair.
A hajszíne / a haja színe - his/her hair color / the color of his/her hair.
We're lazy, sometimes we like to cut off the possesive suffix by gluing the two words into one. Other examples: szemszín, könyvborító (the cover of the book), asztalfiók (the drawer of a desk), óvónőszárny (the wing of a kindergrten teacher. I've just made it up, so you can see how it works. Those flying kg.teachers must get around somehow!)
The point is, that the detached version (a haj színe) is a simple possessive expression, the non-detached is the double-possesive (where hair has a color ad he/she has hair) that you're asking.