Hi,
Just to add something from personal experience.
My great grandmother does not have her father's name of her birth certificate. She was born in 1898. Her parents didn't marry until around 1.5 months after her birth. After her parent's marriage, she adopted her father's surname on census records. She also married using her father's surname, but she recorded the birth of her son using her mother's surname.
A DNA test has shown that the man not listed on her birth certificate was indeed her biological father. As you may not be able to confirm the relationship via DNA testing, it is worth considering the age of the mother when the child was born. The mother's surname may also match with the mother's father at the time of the child's birth. If so, this would suggest that she was not previously married when the child was born.
All the best,