How can a person be born to six fathers?

Well, as movie suggested, it was possible because of genetic engineering.

Scientists did an experiment in genetic laboratory for this. They combined the DNA of six fathers to get a perfect child born.

As per plot from Wikipedia

Julius Benedict and Vincent Benedict are fraternal twins, the result of a secret experiment carried out at a genetics laboratory to combine the DNA of six fathers to produce the perfect child. To the surprise of the scientists, the embryo split and twins were born.

I couldn't find regarding multiple fathers IRL except this one, but it refers to twins with DNA of different fathers. This is a movie, not real life and everything is possible in a movie.

Its been a while, but the way I remember that movie it was multiple prospective sperm donors (six?). If sperm from all 6 were used, it would essentially be a crapshoot which one actually managed to merge with the egg. Without genetic testing (this movie was made in the 80's when that was new and expensive) it would be difficult to tell which one "won".

The additional wrinkle was that there were twins. As Fraternal (not identical) twins, that means two different eggs merged with two different sperm. So it would be possible in that scenario for twins to have different fathers. Technically these twins would be half-siblings. Again, it would be tough in the days before cheap genetic testing to figure out for sure which of the 6 was your biological father. But if you went to visit all the donors, you could perhaps make some educated guesses. So that's what they did.

At least that's what I told myself while watching it. According to the Wikipedia blub they somehow spliced material from 6 donors. That would be scary madness even today. We don't know enough about how multiple genes interact to confidently create a working human that way. It also claims the twins happened when the embrio split. That can happen, but if it does both copies have the same genes, so you get identical twins. So clearly that didn't happen.

So basically the given science for the movie is complete nonsense. I'd advise either ignoring it, or using something akin to my rationalization so you can get on with enjoying the story (which was kind of fun).

Normal human cells have 23 pairs of chromosomes, one of each pair from each parent. An embryo with 'six fathers' might consist of an egg fertilized by a man's sperm which had 20 of its chromosomes removed and replaced with the corresponding chromosomes from five other men, perhaps four chromosomes from each. In that case the genetic material would consist of 23 chromosomes from the mother, 3 chromosomes from the first father, and 4 chromosomes from each of five other fathers, along with chromosomes for the mitochondria which would come from the mother.