The vast majority of the Schemes in the test suite return #f to both (eq? (vector) (vector)) and (eq? (string) (string)). I also tested bytevectors in those implementations that made it easy to do so. Here are the cases that returned #t:
Chez returns #t for both empty strings and empty vectors.
Icarus (but not Vicare) and Chibi return #t for empty vectors only.
Ypsilon returns #t for all three cases.
NexJ returns #t for empty vectors but not empty strings.