I could see her saying that and then giggling and saying oops.

It's not really a mistake, since Fa is her name.

But it would be a mistake to think Mulan was her family name. That mistake might be made by Tiana or Cinderella or Rapunzel. The others seem to come from more fantastical settings where no one seems to have family names at all.

Cinderella seems to come from a multi-cultural region, since the Duke attempts to address her in several languages--even though he knows she's from the local kingdom. So, I think she'd be less likely to assume something about how names are structured.

Rapunzel seems pretty naive so she wouldn't make assumptions. Neither she nor Mother Gothel seem to have a family name.

Tiana, though, is from the 1920's Louisiana, where all names follow a standard first name last name structure and convention.

Because only Mulan is the Southeast Asian.
For example: I'm the Southeast Asian (Vietnamese), my full name is Pham Thi Phuong Linh, others call me "Linh" or "Phuong Linh" not "Pham".

I think she would for sure. Since she was a mermaid, she'd probably be struggling with unusual rules in the human world. Add to the fact that she's a bit of a ditz, I imagine her saying Fa and then giggling about it once she realizes her mistake. I could plausibly see Snow White and Rapunzel making the mistake too, but they didn't come to mind as quickly as Ariel did for this situation.

Because Tiana is from a time where they say their given name before their family name. At least in America.

I doubt any of them's been to Asia before. I actually think Tiana's the only one that might know the difference since travel between different nations was more common then and Asian influence furniture etc. had been popular only a few decades earlier. She also has direct media from film and radio.