I've seen the term "signature monster" thrown around a lot but I'm not exactly following it. What makes these monsters signature? What makes any monster signature to a given game? Is D&D the only game with signature monsters because of its age? And where can I find a list?
"Signature monster" just means a monster that is iconically associated with the game (signature ~= iconic). Not necessarily because of legal/IP restrictions, but because it is either unique to a game, uniquely imagined for that game, or just so popular in that game that there's no question.
D&D has a batch, but there are others: Space Orks with Warhammer 40k, Cthulhu etc. with Call of Cthulhu and its variants, Arachnids with Starship Troopers... Insane melon-headed goblins with Pathfinder... Those are all good examples of monsters that aren't restricted to one game but are signature to one. (IP restriction is another thing; Zergs are signature to Starcraft mainly because they'd sue the bejeezus out of anyone else using them.)
There is no "official list," it's a look and feel thing. For D&D, most people would cite mind flayers, carrion crawlers, displacer beasts - popular stuff that's been around for 30 years and if you pulled out a picture people would say "D&D!"
Signature Monsters: The monsters that readily call to mind certain games, due to being unique to (or almost unique to) that game, or at least treated in a new and unique manner.
For D&D, those are Illithids (Mind Flayers), Beholders, Umber Hulks, Owlbears and Rust Monsters come to mind.
For the D&D Dark Sun setting, the Thri-kreen.
For Warhammer Fantasy Battles and FRP, it's the Greenskin races as a set Orks, Goblins/Gretchin, Squigs, Trolls. These are ones where GW went WAY afield of the mythology. And the Skaven.
For Warhammer 40K: Space Marines, Genestealers, Tyranids, Squats. And Squats haven't been part of the game for over a decade now... but still, you say squats, many gamers think "Biker-Viking Space Dwarves"...
Traveller has the Droyne and Hivers... these two aliens are visually and culturally distinctive, and seeing either immediately calls to mind Traveller for most fans of the game.
For the Fantasy Trip, the Barrow-Wight, and Children. Only RPG I've seen to list children in the bestiary (under the heading of Nuisance Creatures).
GURPS Fantasy has blue Kobolds...
The grinning nasty dog-with-mouth-like-shark kobolds of Kobolds Ate My Baby are distinctive. So is most of Kovalic's art, but that one is an instant Rorshack Test for "KAMB player."
Hobbits for LOTR. Vulcans, Orions, Romulans and Klingons for old Trek.
For Red Dwarf (both series and RPG), the Curry.
For Judge Dredd, you get Judges, Fatties, and the Dark Judges.
Orkworld has it's Orks, Shtuntee, Ahlvesees, Shtoontee, and Traals... not so unique, but all treated uniquely.