I do not think people are specifically born with the talent to sing. There are some out there that are completely tone deaf; no matter how hard their fame hungry parents force them to attend singing lessons or the like throughout their childhood years, they will never be good singers. On the other hand if you have an aptitude for singing this can be honed and your voice can be trained to sound its ultimate best.
Can you imagine the world never having had the pleasure of the voices of singers such as Frank Sinatra, Ella Fitzgerald, Michael Jackson, Asha Bhosle, Whitney Houston, George Michael, Mariah Carey and Usher to name just a few? You might not agree that the aforementioned have/had great singing voices but they had a talent which made millions of people buy their music over the years.
Then you have people like Robbie Williams, Britney Spears, Shakira, etc, who I personally don't think have particularly wonderful singing voices but are natural entertainers who make money from their ability to charm people into thinking they are good singers!
I have friends whose parents made them attend singing lessons when they were growing up and some of them are really scary when they sing at family parties and should really not be allowed to sing in public. On the other hand I know people who seem to have a natural flair for singing and just sing at family events when really pushed. Isn't it funny how the people who seem to be tone deaf clamour for the chance to sing in public and those who are actually very good are the ones who don't push themselves forward?
To summarise I would say that great singers are born with some degree of talent which has to be nurtured over time to develop into something so wonderful that people are in awe of their voices. I'm very glad my parents didn't make me attend singing lessons as I grew up; as the only place my singing should be heard is in the bathroom with no other humans within earshot!