There is a canon discrepancy regarding the Invisibility Cloak. You are correct that Harry's cloak was described as a "true" Invisibility Cloak, immune to charms and spells, indestructible, and invisible even to Death, by Xenophilius Lovegood.
However, in Tales of Beedle the Bard, Dumbledore's notes represent the cloak differently and not quite as impervious as Xenophilius Lovegood claimed. In Tales of Beedle the Bard, the cloak is described as having a "uniquely durable nature" and is not described as indestructible, impervious, or immune to Death's gaze. J.K. Rowling writes:
Invisibility Cloaks are not, generally, infallible. They may rip or grow opaque with age, of [sic] the charms placed upon them may wear off, or be countered by charms of revealment. This is why witches and wizards usually turn, in the first instance, to Disillusionment Charms for self-camouflage or concealment.
Tales of Beedle the Bard - pages 96-97 - Bloomsbury Edition - chapter 5, The Tale of the Three Brothers
We know for certain that at least one spell works against the Invisibility Cloak: Homenum Revelio. In Chamber of Secrets, Dumbledore uses Homenum Revelio to see Harry and Ron under the Invisibility Cloak in Hagrid's hut:
‘However,’ said Dumbledore, speaking very slowly and clearly, so that none of them could miss a word, ‘you will find that I will only truly have left this school when none here are loyal to me. You will also find that help will always be given at Hogwarts to those who ask for it.’
For a second, Harry was almost sure Dumbledore’s eyes flickered towards the corner where he and Ron stood hidden.
Chamber of Secrets - page 195 - UK Hardcover - chapter 14, Cornelius Fudge
J.K. Rowling confirms this in an interview:
Angela Morrissey: Why is it that albus dumbledore can see harry under his invisibility cloak at certain moments? (during the series is the cloak only infallible to those who do not own a deathly hallow).
J.K. Rowling: Dumbledore, who could perform magic without needing to say the incantation aloud, was using ‘homenum revelio’ - the human-presence-revealing spell Hermione makes use of in Deathly Hallows.
J.K. Rowling Webchat - 07.30.07 - [THE LEAKY CAULDRON]
If Hermione could master Homenum Revelio, I'd wager Barty Crouch Jr as Mad-Eye Moody could have as well. Barty Crouch Jr was a bad guy and a Death Eater, but he did demonstrate he was competent at magic throughout Goblet of Fire. So Homenum Revelio is one possibility for how Crouch Jr/Moody was able to see Harry through the Invisibility Cloak. I interpret canon, though, as Mad-Eye's eye having powers above and beyond Homenum Revelio. Crouch Jr/Moody actually interacted with Harry while Harry was under the Invisibility Cloak; they had a silent conversation. It's clear Crouch Jr/Moody saw Harry. It's not clear whether Dumbledore saw Harry and Ron under the cloak in Hagrid's hut, or if he merely detected human presence (which is the extent of Homenum Revelio's power), then discerned what general area it was coming from, and knew it was Harry because who else (that we know from canon, circa Chamber of Secrets) had an Invisibility Cloak at Hogwarts?
Putting it lightly, it's likely Moody's magical eye was charmed to have exceptional and/or rare sight abilities -- heck, Moody probably could've seen the future with that eye! Okay, not really, but at the very least the magical eye was a prosthetic device that Moody could effectively channel revealing charms through (although that Ollivander in Deathly Hallows says a wand is the required instrument for channeling magic should be taken into consideration. If it actually channeled magic, I believe Moody's eye would be the only other instrument besides a wand to channel magic in canon¹). At most, the eye itself is a very powerful magical object (this is what I tend to think) which has magical abilities that just aren't directly explained by canon. We can only infer. Another point to remember is Dolores Umbridge was able to fasten the eye to her office door and presumably use it to channel sight in some way; we don't know from canon exactly how Umbridge used the eye, although it seems she merely used it to see who was approaching her office.
While a powerful magical object, I don't see Moody's eye as being on the same level as the Hallows.
¹I realize that some people parse Ollivander's words to mean a witch or wizard can channel magic through any kind of instrument -- meaning anything, any object -- but in the context of the conversation, I interpret Ollivander's words to mean a wand is the singular instrument for channeling magic and that almost any wand will work for a magical person. Deathly Hallows - chapter 24 - The Wandmaker - page 399 (UK) - page 494 (US)