The answer to this question is going to depend on how "metaphorical" you feel the novel is. We can say with certainty that, for Billy, the Tralfamadorians are real. His time travel is real. Everything included in the novel is presented as real/factual. These are not moments of fantasy. Every moment (after Chapter One) is part of the same reality.
The reality of the novel is Billy's reality.
Critic Robert Scholes commented on the absurd elements of the novel.
He asserted that the absurd elements of the novel are appropriate and necessary to deal with the absurdity of the world. (eNotes)
Taken as an integral part of the novel's program and commentary, the alien elements and science-fiction portions of the story are indeed "real".
While Billy's situation is literal in the sense of being a science fiction device—he is ‘‘literally’’ travelling through time—it also serves as a metaphor for the sense of alienation and dislocation which follows the experience of catastrophic violence (World War II). (eNotes)
Some readers like to view the novel differently. These readers will point to the fact that Billy only begins to tell people about the Tralfamadorians after he has been in a plane crash, suffered a brain injury and lost his wife.
Trauma is the explanation for Billy's science-fiction types of experiences for these readers.