If someone has no way to tell who a person's parents are/were (no paperwork), is there a way to tell for a Wizard if someone is a Squib or a Muggle? Or are they 100% functionally equivalent and only different as far as bureaucracy is concerned?
You could perform what I'm going to call "the Dementor test". As we learn in Order of the Phoenix, Dementors are invisible to Muggles, but can be seen by Squibs. If you really needed to, putting them in front of a Dementor and checking if they can see it would be one way to test whether they're a Squib or a plain old Muggle.
Other than that, Squibs seem to be able to see things that Muggles can't. They're not affected by charms and spells intended to disguise objects/locations or repel Muggles, so Argus Filch can see Hogwarts as it really is, rather than as an old ruined castle.
I would rather say - "No" or at least there is not direct test that wizards are aware of and that can distinguish between a muggle and a squib (except for checking their parents).
If we check following dialogue from Harry Potter and The Order of the Phoenix, chapter 8:
'I'm a Squib,' said Mrs Figg. 'So you wouldn't have me registered, would you?'
'A Squib, eh?' said Fudge, eyeing her closely. 'We'll be checking that. You'll leave details of your parentage with my assistant Weasley. Incidentally, can Squibs see Dementors?' he added, looking left and right along the bench.
This implies several things:
- Wizards do not have a quick way (like a spell or so) to tell if someone is a squib - Fudge needs to order a background check for it.
- Also most likely there is no a slower test that can confirm this - Fudge does not say "we'll do the squib test later to confirm your statement". Also Dumbledore does not offer to perform such a test so that the authenticity of his witness is proven.
- Wizards have a little knowledge of the skills that squibs possess - Fudge is not sure whether squibs have the ability to see Dementors.
So most probably the wizards don't know a way to determine whether a person is a squib or a muggle. It is possible that such a test exists in the form of a spell or a potion but since wizards don't care much for squibs, they don't care also much to develop them.
Things like being able to see a ghost or a Dementor could serve only as positive confirmation. For example a squib that does not want to be identified as such might just lie that he/she does not see a ghost when asked.
Another possible way would be to directly ask the question and use legimency to check whether the person is telling the truth. Still this also has some issues :
- It is very intrusive so it cannot be used in any situation. Fudge for example did not dare using it of Arabella Figg.
- Very few wizards are adept at it.
- Legimency is not like reading one's mind. The asked person might just ignore the question. In this case the wizard cannot determine whether someone is telling the truth since no answer is given.
You can perform the Moldering Old Ruin Test
When Muggles look at Hogwarts, all that they see is a moldering old ruin.
“But Hogwarts is hidden,” said Hermione, in surprise. “Everyone knows that . . . well, everyone who’s read Hogwarts, A History, anyway.”
“Just you, then,” said Ron. “So go on — how d’you hide a place like Hogwarts?”
“It’s bewitched,” said Hermione. “If a Muggle looks at it, all they see is a moldering old ruin with a sign over the entrance saying danger, do not enter, unsafe.”
Harry Potter and the Goblet of Fire - Chapter Eleven
However, we see that squibs have no problems entering Hogwarts.
“He did it, he did it!” Filch spat, his pouchy face purpling. “You saw what he wrote on the wall! He found — in my office — he knows I’m a — I’m a —” Filch’s face worked horribly. “He knows I’m a Squib!” he finished.
Harry Potter and the Chamber of Secrets - Chapter Nine (text legally available online)
Angus got as far as the Sorting Hat before he was exposed. In sheer desperation he threw himself ahead of a girl whose name had been called and placed the Hat upon his head. The horror of the moment when the Hat announced kindly that the boy beneath it was a good-hearted chap, but no wizard, would never be forgotten by those who witnessed it. Angus took off the hat and left the hall with tears streaming down his face.
Presumably in order to walk around Hogwarts, they were able to actually see it. Thus squibs can see Hogwarts while muggles cannot. Bring the unidentified person to Hogwarts and ask them what they see.
From Pottermore we get this quote on Muggles and their inability to see ghosts:
Witches and wizards are much more susceptible to what Muggles call paranormal activity, and will see (and hear) ghosts plainly where a Muggle might only feel that a haunted place is cold or 'creepy'. Muggles who insist that they see ghosts in perfect focus are either a) lying or b) wizards showing off - and in flagrant breach of the International Statute of Secrecy.
Now we know that muggles cannot see ghosts but do we know that squibs can?
I think we can take directly from canon that they can definitely see poltergeists, this is evidenced multiple times through Argus Filch and Peeves. I'm fairly certain Filch has interacted with a ghost on at least one occasion, leading me to suggest that you can use a much safer means of testing Squibs and Muggles by using ghosts.
(Unfortunately I don't have my books to hand at the moment)
The Harry Potter Wikia has a whole section with the subheading "Differences between Squibs and Muggles." Some might say that is answer enough, but I'll go deeper.
In this section, which can be read in its entirety here, is the explanation:
"In addition to being born into magical rather than Muggle families, Squibs are different from Muggles by their knowledge of the wizarding world. They are able to see things that are hidden from the Muggles and have access to certain magical objects and creatures that can help them. Argus Filch, as well as Arabella Figg, have formed wizarding-like relationships with cats."
Others have spoken about their ability to see ghosts and Dementors. Additonally, the relationship between Mr Filch and Mrs Norris is something I had not originally thought of, but I think it is very clear that there is something magical about their relationship and connection. Whenever there is a problem, Mrs Norris seems to come straight to Filch or he comes straight to her. There is a popular rumour that Filch was at least part Kneazle, like Crookshanks, which would make her a magical creature, however this has been denied by JK Rowling on Twitter. I'm not suggesting that Filch really is magical, but perhaps their relationship has grown to that extent because of some residual, non-Muggle magical potential.
I think that the ability to see things a Muggle cannot and special situations like Filch's relationship with his cat serve as evidence that a Squib has magical potential and will likely have had an upbringing that makes them different to a Muggle, who have zero magical sensitivity, potential or ability. Filch obviously thought so too since he ordered his KwikSpell course in an attempt to develop his magical skills. Of course this didn't work, and the KwikSpell company are obviously taking advantage of Squibs, but it seems to me further evidence that there are people who would accept the idea that a Squib certainly has more magic in them than a standard, nothing-to-do-with-the-wizarding-world Muggle.