The only reason why Hatake Kakashi keeps wearing the mask is to hide his malformation due to not controlling the dog sage mode properly. His father the white fang was renown for being equivalent in talent to a sannin hence his power to have a pact with dogs. During times of war, the white fang felt probably the need to initiate his son early to the power of the sage mode, or did Kakashi sneak on him and train behind his back.
In any case, things went bad for Kakashi when he couldn't control the sage mode and transformed part of his face into that of a dog. The reason why the dogs all remember him differently is that, being of different breeds each conceives his looks based on his point of view.
The white fang could enter the Sage mode, hence his name the white fang.
Just my thoughts...