She didn't expect other people to experience the object the way she did.

现代英语，或者说英语发展到今天，the way 已经成为名副其实的“连词” 了。本句，the way = as（引导方式状语从句）

She didn't expect other people to experience the object as she did.

did 是个替代词，等于 experienced the object。