Good observation! The Buddha himself emerged from Hinduism, didn't he? Were it not for being a Hindu, he might never have become realized as the Buddha.
Vivekenanda's statement is not simply a reflection of his thoughts or beliefs, but an insight into the nature of Reality. If you take the time to actually read and absorb this very cogent statement, you will SEE the veracity of his insight, and realize that it is NOT based upon belief or some superficial passing thought or conjecture. In fact, the statement itself reveals how we see Reality via conceptual thought, as compared to seeing things as they actually are, non-conceptually. Here is the point at which what Vivekenanda sees and what the Buddha sees are one and the same Reality. How can it be any different, if both are seeing correctly?
You consistently attack the pointing finger, rather than to have a close look at that which is pointed to. Are you ready for a shift in consciousness, or do you still prefer the delusions brought about by the dancing cave wall shadows?