I don't know if you accept it as normal, but I've seen that in each of the four companies I've worked in. Sometimes the chief of staff at the VP level is not an engineer, but the chief of staff to CxO I've seen are usually engineers.

speaking from experience at Google, yes. Obviously depends on size of company, but many C-suite/SVP/High VPs would have a "chief of staff" type person reporting to them.