Why does the Foe-Glass work?

Question

In Harry Potter and the Goblet of Fire we are introduced to Moody. He has an assortment of Dark Detectors. One of these dark detectors is the "Foe-Glass". How does the Foe-Glass choose whose foes to display? Harry visits Moody before the first task and Moody says "See them out there, skulking around? I'm not really in trouble until I see the whites of their eyes". This implies that the Foe-Glass is currently reflecting Moody's foes - not Harry's - so it doesn't just display the foes of those that look into the glass. This makes sense if the Foe-Glass belongs to Moody.

However, (GoF spoilers)

The person that is pretending to Professor Moody is actually Barty Crouch Jr. He even states that the dark detectors were stolen from the real Moody so this implies that the Foe-Glass was actually the real Moody's. Why would it then seem to display the foes of Barty Crouch instead of Moody? Does it somehow 'know' that the impostor currently possesses it - similar to how the elder wand 'knows' who its rightful owner is?

How does the Foe-Glass decide whose foes to display?

Answer 1

It seems that the Foe-Glass shows the enemies of the person who physically possesses it, and doesn’t seem to distinguish this from its “true” owner.

It’s definitely Crouch’s foes who are appearing in the Foe-Glass, as shown when his identity is revealed at the end of the book:

Harry, still staring at the place where Moody’s face had been, saw Albus Dumbledore, Professor Snape, and Professor McGonagall looking back at him out of the Foe-Glass.

— Goblet of Fire, chapter 35, Veritaserum

There may be some sort of enchantment to “claim” the Glass (which is why Harry doesn’t see his enemies when he looks in the Glass, for example), but details are never given. But it certainly isn’t as sophisticated as something like, say, a wand.

Answer 2

I imagine the Foe-Glass to be like the mirror of Erised. Only the user's desires, or in the Foe-Glass's case enemies, are displayed.

Here is a quote that might explain a bit, from the Harry Potter wikia:

Barty Crouch Jr., while impersonating Alastor Moody, had Moody's Foe-Glass in his office during Harry Potter's fourth year at Hogwarts School of Witchcraft and Wizardry. He said he's not really "in trouble" until he can see "the whites of their eyes."[1] Later in the school year, after Cedric Diggory's tragic death, it was revealed that Crouch's enemies shown in the glass were Albus Dumbledore, Minerva McGonagall, and Severus Snape, all notable members of the Order of the Phoenix.

So this could mean, if this is reliable, that the true owner's enemies are displayed. How does it know? Possibly the same way a wand knows: something magical.

And in this quote from a Wikipedia page on Magical Objects.

A Foe-glass is a mirror that detects and shows its owner's enemies in or out of focus, depending on how close they are. However, like all dark detectors, it can be fooled, as mentioned by Harry in the fifth book at the beginning of the first D.A. meeting. Moody, in reality Barty Crouch, Jr. in disguise, claimed that when the whites of their eyes are visible, he is in trouble. When Moody/Crouch was attacking Harry, he watched Professors Snape, McGonagall and Dumbledore approach the room before they actually got there in the mirror. A Foe-glass is hanging in the Room of Requirement in Harry Potter and the Order of the Phoenix when Harry uses the Room for D.A. meetings.

And in this answer from Wiki Answers, it says that the enemies displayed in the glass were Dumbledore, Snape and McGonagall. This matches the answers and these enemies were only for Barty, not the real Mad-Eye.

So the foe glass only displays the enemies of its owner.