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Martians can totally count as monsters! Marvin the Martian and the titular alien soldier are both legit.
Conversely I'm saying that ToeJam & Earl are not mosnters. Too friendly, not on Earth with any ill-intent. I'd be interested in focusing on what makes games with monsters different from more standard protagonists. Not everyone needs to look for that difference, but it gives a little more meaning to the theme!

Here's a list I have going of some more monster-game choices:
Bloodrayne (Gamecube/PS2/oXbox) - dhampir
Castlevania: Symphony of the Night (PS1/PSP/XBLA/PSN) - dhampir
Darkstalkers 3/Vampire Savior (Arcade/PS1/Saturn/XBLA/PSN) - various
Dragon Quest Heroes: Monster Slime (DS) - slime
Gargoyle's Quest (Game Boy/3DS Virtual Console) - gargoyle
Geist (Gamecube) - ghost
Godzilla: Destroy All Monsters Melee (Gamecube/Xbox) - various giant monsters
Kid Dracula (Famicom/Game Boy) - Vampire
The Legend of Zelda: Twilight Princess (Gamecube Wii) - werewolf
Monster in My Pocket (NES) - vampire/Frankenstein's monster
Rampage (Arcade/NES/SMS/too many to list) - Giant Ape/Lizard/Wolfman
Toxic Crusaders (NES/Game Boy/Genesis) - Mutant/Wolfman
Vampire: The Masquerade - Bloodlines (PC) - vampire
Werewolf: The Last Warrior (NES) - werewolf

It's always a good time for Splatterhouse.

Additional monster games to check out!

Alien vs. Predator (SNES) - Predator
Aliens versus Predator (PC) - Alien/Predator
Bloody Roar series (PS1/PS2/GC/Xbox) - Zoanthropes
Destroy All Humans! (PS2/Xbox) - Alien
Giants: Citizen Kabuto (PS2/PC) - Aliens/giant monster
King of the Monsters (MVS/SNES/Genesis) - Various giant monsters
King of the Monsters 2 (MVS/SNES/Genesis) - Various giant monsters
Legacy of Kain series (PS1/PS2/Xbox/PC/DC) - Vampires
Wolfchild (SNES/Genesis) - Werewolf or Wolf/Human hybrid

And don't forget the variety of fighting games with monster characters: Mortal Kombat, Killer Instinct, Soul Calibur, MACE: The Dark Ages, Primal Rage, etc.

I keep forgetting I have Giants to play through. I can definitely recommend Legacy of Kain: Soul Reaver to anyone looking for something to play.

I keep forgetting I have the first Legacy of Kain to play through! I totally forgot I picked it up on a PSN sale AGES ago~
Finally a time to go back and finish that game. Dipped into it a little on the original hardware quite a few years ago, but never got back to it despite enjoying it. No better excuse than next month ^w^

A good reason to finally getting around to playing Destroy All Humans!

Hey! #notallhumans

It's the only way to be sure.