As has been pointed out in the comment above, the specific claim that we see ourselves as exactly five times more attractive is a little absurd. There are no units measuring beauty, that I am aware of.
In social sciences, when the attractiveness of an individual has to be rated, it is rated on a scale where, for example, one would mean low attractiveness and five high attractiveness. Under such scales, everyone in the study would literally have to be of a 5/5 attractiveness and yet rate themselves as a 1/5 attractiveness. That's highly unlikely, to say the least. Thus, I'm inclined to dismiss the notion that we see ourselves five times more attractive than we are as bogus.
Leaving at that, however, is leaving much unanswered. There is, I think, a vaguer but still quite interesting claim to investigate: Do we overestimate our own attractiveness?
That question has an answer, and the answer is yes.
According to the authors of The Invisible Gorilla: And Other Ways Our Intuitions Deceive Us, relating the findings of Narcissistic Illusions in Self-Evaluations of Intelligence and Attractiveness,
Evidence about self-judged attractiveness comes from a study of college students in which men judged themselves to be about 15 percent more attractive than they actually were. Women viewed themselves as slightly less attractive than they actually were, although both men and women viewed themselves as above average in attractiveness (the women in the study were judged to be a little more above average in attractiveness).
The study itself is unfortunately locked behind a paywall, but the abstract does support this:
Both males and females overestimated their own intelligence, with males, but not females, also overestimating their attractiveness. ... Males showed greater positive illusions than females, with this effect at least partly attributable to observed gender differences in narcissism
The theory that we overestimate our own beauty is supported by another study: Mirror, Mirror on the Wall: Enhancement in Self-Recognition.
Researchers took photos of undergraduate students and then invited those students back to the laboratory two to four weeks later to identify their face out of an assortment of eleven images. Out of those eleven pictures, five were altered to make the subject less attractive than the original and five were altered to make the subject more attractive. The eleventh one was the original, unaltered picture
The result? The majority of participants picked one of the pictures that was altered to be more attractive than the original picture, overestimating the attractiveness by six to twelve percent.
The researchers then did a similar experiment, but this time they asked the participants to evaluate friends and strangers, in order to see if we also overestimate other people's attractiveness or only our own. In the end, subjects overestimated their friend's attractiveness but not the stranger's.
TL;DR: We do overestimate our own beauty, but not by anything close to a magnitude of five times.