Is it normal to be turned on by the feel of fur?

Very Normal.
The first toys we are given are teddy bears and other stuffed animals as babies. For some of us, we keep these creatures even beyond our infancy and into adulthood.
The stuffed animal gives us comfort, security and when we feel the feel of the fur, it reminds us of this.
This comfort may even be felt by petting animals and remembering petting our most beloved pets.
This comfort is something safe and feels soft and kind.

Where it becomes something that is a fetish is not something I can easily comment upon. As a furry who lives with a girlfriend who is furry, for her and I, it seems that when we wear our costumes we remind the other of a large, breathing, "living" stuffed animal. Petting the fur is something I also enjoy.
It is not really new either. It has been documented as a fetish and has been the subject of literature (Steinbeck's Of Mice and Men) Lenny, a character, was fascinated by petting soft furry little creatures.

The soft feeling may just be comforting and you may just like the texture against your skin :3