Amity park citizens, for the most part, think that Dani is Danny Phantom's little sister. Sure there are the few like Vlad and Valerie that know the truth but everyone else ran with the idea that those two half ghosts were siblings.
It wasn't just the fact that those two looked alike or had similar colored outfits with the same symbol. It was mannerisms, them spending time together, and how protective Danny was to the younger halfa.
Dani's first debut to the public eye had been her chasing her cousin, covered in a mysterious goo, demanding that he stop and give her a hug. The news only caught wind of it because the chase lasted from the middle of town to the edge of it and then back to the otherside.
Most bystanders were worried that the goo had some sort of unknown side effect. Val and some of Danny's fans had come out to run interference. Their help led to Danny being goo-ifed, when he had to snap at someone (not Val) to not shoot his cousin, proving that the goo was harmless. It was gross and hard to clean, but harmless nonetheless.
Danny's only retaliation for being slimed, was a quick head lock and knogey that lasted no more than 5 seconds before telling her to go home and get clean.
The next day the two Phantoms were spotted getting ice cream before making a mess by trying to finger paint each other's faces with frozen dairy.
Then there was the time the MIW captured Dani. The MIW had to learn the hard way that Mr. Phantom had been going easy on them in previous fights when they refused to release her. Their headquarters and three other buildings had become a pile of ruins during Danny's quest to find his cousin.
No news reporter could get an interview as to why it happened that way, but someone reported that they overheard Phantom say he wasn't going to just let someone experiment on a member of his family.
That was the first time the idea of Phantom having a family was brought to the public's attention. It was then from that moment on that Dani was assumed as Danny's little sister.
Danny for his part didn't know of this assumption until someone had managed to ask him where his sister was. He was fairly sleep deprived -school plus late night ghost hunting- at that moment and, thinking of Jazz, casually replied that she was at home with Mom and Dad. It was only later, when he was thinking more clearly, did he have a mini panic attack over some random knowing he had a sister, which had to mean his secret was out. His panic lasted till he found out about the assumption of his clone.
Dani found out when she tried to correct someone on it, only for Danny to stop her saying she's been upgraded to baby sis.