Of course you have the right to ask your mother. This person has entered your space and world and you are absolutely entitled to inquire. Talking to your mom is 100% more respectful to her than going around her back and hiring a PI or etc.
And to some extent I think you have some rights to know if you have a sibling. A sibling can be a special thing and more concretely a sibling can provide medical help in the form of kidney or bone marrow or familial history clues of disease (especially true for something like breast cancer or cancer in general).
However. I don't think your right to know trumps your mom's right to privacy on this matter. She is entitled to keep this a secret if she wants. That's why if you inquire and she vehemently denies you need to drop it. And your mom could be telling the legit truth. Nothing you have said makes me think this person's thoughts on her parentage are infallible.
And FWIW, I see nothing wrong with asking your mother once in a respectful, non confrontational manner, provided it's with the intent of letting her know you are there for her if there's something she would like to get it off her chest and offering her a non-judgmental outlet. It should not be so that you can get to the bottom of it or find your own sense of closure. My comments were more focused on this idea that you have a "right" to know that trumps her right to privacy, and that that right gives you not just free reign to question her but go digging around in her past somehow to get to the bottom of it.