The reason Freud called his analysis "scientific" is for rhetorical purposes. He believed that science had given the lie to all things "supernatural", but the interpretations of dreams was still full of supernatural claims, which he hoped to eliminate through his own set of materialist rules. As he says in The Interpretation of Dreams (1913):
It would ... be wrong to suppose that the theory of the
supernatural origin of dreams lacks followers in our own day; for
leaving out of consideration all bigoted and mystical authors—who are
perfectly justified in adhering to the remnants of the once extensive
realm of the supernatural until they have been swept away by
scientific explanation—one meets even sagacious men averse to anything
adventurous, who go so far as to base their religious belief in the
existence and co-operation of superhuman forces on the
inexplicableness of the dream manifestations.
Everyday life in 20th century Europe lacked religious experiences, but for example, people still saw Jesus in their dreams, and considered this to be a proof that they could be visited by divine agencies. If Freud could "prove" that Jesus was merely a manifestation of their minds, he could prove that dreams were no more supernatural than any other human experience, and this would be a more "scientific" claim than the commonly held beliefs of his time.
21st century scientists would say that Freud and Jung were doing subjective analysis, not neuroscience. By Popperian standards, Freudian analysis is not scientific. (source) This does not mean it's useless, just that it's not much different from Freud's hated "supernatural" interpretations.
There is a scientific method of dream interpretation, the cognitive neuroscience of dreams, but it has only made very basic findings, such as the difference between REM and NREM dreams.
