How come I have rotator cuff syndrome, but nobody else in the family has it?

Not inherited. Rotator cuff syndrome develops from the way you use the 4 muscles that stabilize the humerus (long arm bone above the elbow). Although alignment of your clavicle (which may be inherited) can contribute, more common causes include muscle imbalance and poor shoulder mechanics. Many people have tight chest muscles, out of proportion to back muscle strength. A physical therapist can help correct this.
No genetic linkage. There is no strong family linkage with rotator cuff tendinitis. If you experience symptoms depends on the interplay of how your shoulder is built/functions and the how you use your shoulder. While clearly you are built in a fashion that is similar to other family members, it is not out of the ordinary that not everyone has shoulder problems (yet).
Not familial. It is not a familial problem, but usually related to injury or overuse.
Bad Luck? Or even could be due to activity you do. Hard to say without more info. See a musculoskeletal specialist or orthopedic surgeon to discuss your options for care.