All Pets:
Format:
Item Name - Pet name - Source (Anime, Game, etc.)
Sorted by the game's sort button. (literally)

Blood Crystal - Flandre - Touhou
Friendship Cookie - Cultist - Enter the Gungeon
Ancient Helmet - The Heroine (Undyne) - Undertale
Ancient Katana - Youmu Konpaku - Touhou
Dark Flag - Bowser - Mario Franchise (obviously)
Box of Bones - Papyrus - Undertale
Energy Heart - X - Megaman X
Eye in a Glass Jar - Eye Rol - Yrimir.
Ancient Mask - Meta Knight - Kirby
Apple - Kid - I wanna be the guy
Black Bell - Mr. Game and Watch - Ditto (Note: Makes an old phone ringing sound on equipping)
Blood - Ice Bat - Terraria Itself
Blood Dye - Giant Bat - Terraria Itself (again)
Blood in a Flask - Vampire Bat - TI (And again)
Blood in a jar - Lava Bat - TI (and another one)
Blood in a potion - Illuminant Pet - TI (And another one!) [Not to be confused with "illuminati" bat]
Blood Soup - Jungle Bat - TI (And another one!!)
Bloody Bass - Flesh Reaver - TI (And Another one!!!)
Blue Etherian Mana - Etherian Wyvern - TI (AND AN..... it's not blood so, i'll let it pass)
Blue Feather - Blue Jay - TI
Crystal Bass - Crystal Thresher - TI
Dark Crystal Ball - Wraith - TI
Dark Gel - Slimer - TI
Desert Core - No-Internet Dinosaur - Google Chrome's no internet page.
Dog Cookie - Pug - It's a pug, cmon.
Dried Bass - Sand Shark - TI
Ellen's Diary - Ellen - The Witch's House
Explosive Jam - Creeper - Minecraft
Fruit - Yoshi and Baby Mario - Yoshi's Island
Giant Shell - Giant Tortoise - TI (Note: Doesn't seem to go away even if you equip another pet)
Gohei Stick - Reimu Hakurei - Touhou
Golden Carrot - Golden Rabbit - TI
Fan - Yuyuko Saigyouji - Touhou
Ancient Bell - Cirno - Summon Night
Beef - Kirby - Ditto
Broken Hammer - King DeDeDe - Kirby
Sledge Hammer - Builder Mario - Super Mario Maker
Garlic - Wario - Mario Franchise
Giant Sword - Undyne (again) - Undertale
Golden Bonnie Head - Purple Guy - Five nights at Freddy's
Golden Smash Ball - Master Hand - Super Smash Bros.
Flower Necklace - Alice - Sentry Lore (Check the mod's First page)
Scarlet Orb - Remilia Scarlet - Touhou
Master Ball (#382) - Kyogre - Pokémon
Pokéball (#258) - Mudkip - Pokémon
Metroid Capsule - Metroid - Metroid (Another Metroid just to be sure)
Masterball (#251) - Celebi - Pokémon
Ancient Energy Stabilization Core - Blue Scarab (yeah... i know) - TI
Time Controller Watch - Sakuya Izayoi - Touhou
Shadow Prism - Mudkip Warrior - Pokémon.... i guess.
Gold Feather - Gold Bird - Gold TI
Crystalized Green Etherian Mana - Etherian Wyvern - TI
Green Onion - Hatsune Miku - Touh-.. no wait.
Gross Sack - Floaty Gross - TI
Forge Hammer - Edgar Colthearts - Summon Night
Koopa Shell - Koopa - Mario Franchise
Living Flame - Grillby - Undertale
Magician Hat - Marisa Kirisame - Touhou
Mechanical Key - Baby Skeletron Prime- TI
Special mushroom - Yoshi - Mario Franchise
Overfrozen Crystal - Cirno - Touhou
Cinnamon Pie - Toriel - Undertale
Purple Etherian Mana - Etherian Wyvern - TI
Red Feather - Cardinal - TI
Rotten Bass - Bone Biter - TI
Shy Guy Mask - Shy Guy - Mario Franchise
Magician's Book - Patchouli Knowledge - Touhou
Guide's Autopsy Report - Phoenix Wright - Ace Attorney
Ketchup Bottle - Sans - Undertale
Soul Contract Parchment - Cuphead - Cuphead
Music CD - Napstablook - Undertale
Ancient Mage Hat - Black Mage (The kind of mage not his skin color, ye racist git) - Final Fantasy
Strange Owl Plushie - Security - Five Nights At Freddy's World
Pokéball (#92) - Ghastly - Pokémon
Pokéball (#83) - Farfetch'd - Pokémon
Golden Ring - Sonic (not sanic) - Sonic The Hédge Hoge
Pokéball (#303) - Mawile - Pokémon
Pokéball (#218) - Kirlia - Pokémon
Red Paper - Fujiwara no Mokou -Take a guess... (touhou)
Shangai Doll - Alice Margatroid - Touhou
Pink Umbrella - Yukari Yakumo - Touhou
Portable Witch Effect Egg - Madotsuki (Witch effect) - Yume Nikki
Ten Second Timer - Cute Box - You have 10 Seconds
Portable Knife Effect Egg - Madotsuki (Knife effect) - Yume Nikki
Suspicious Looking Egg - Baby Mothron - TI
Suspicious Looking Cookie - Gingerbread Man - TI
Book of White Magic - White Mage (Not the skin color, again) - Final Fantasy
Yellow Flower - Asriel - Undertale
Strange Monitor - Zero - Megaman X
Triforce of Courage - Link - Legend of Zelda
Stable Core - Gaster - Undertale
Portable Umbrella Effect Egg - Madotsuki (Umbrella effect) - Yume Nikki
Portable Yuki-onna Effect Egg - Madotsuki (Yuki-Onna effect) - Yume Nikki
Portable Demon Effect Egg - Madotsuki (Demon effect) - Yume Nikki
Streetsign - Tricky The clown - Madness Combat
Suspicious-Looking Eyeball - Wegee - Some random dude on the web
Suspicious Looking Snowpile - Greater Dog - Undertale
Tem Flake - Temmie - Undertemmie
Tottally Normal Bone - Annoying Dog - Undertale
Uboa Doll - Poniko - Yume Nikki (Using a light switch while this is equipped might bring out a special someone)
White Feather - Bird - TI
Yume Nikki - Madotsuki (No effect this time) - Yume Nikki
Stack of Five Apples - Kitty White - Hello Kitty
Suspicous Looking Gem - Ryo-ohki - Tenchi Muyo
Bottomless Mud Bucket - Shrek - Shrek
Red Yarn Ball - Bubsy - A crappy game for the playstation 1
Sunglasses - Shades - Enter The Gungeon

Special Pets:
These don't work like the rest of the pets, they cannot be equipped and must be activated manually like before 1.3 They also don't go away when you equip another pet. Both of them can also be out at the same time effectivelly giving you 3 pets at once (4 with the giant tortoise) [these 3 and another one]
Flesh meat - Crimera - TI
Golden Freddy Plushie - Crying Child - Five Nights at Freddy's

Light Pets:
Sigil of Cirno - Sigil of Cirno - Touhou ( i guess)
Heart Container - Heart of Determination - Undertale

Mounts:
Helicopter Key - Apache Chopper - Real Life (Sprite ripped off Metal Slug)
Truck Key - American Truck - Real Life ( I guess kenworth?)
Portable Desert Ant Nest - Antlion Charger - TI
Bloodied Remains - Wolf - TI
Crimson Web with Eggs - Blood Crawler - TI
Blue Tetris Block - 2x2 Square piece - Tetris
Broom - Witch Broom - Every single witch tale ever.
Clown Car Key - Clown Car (Bowser's Flying thingy) - Mario Franchise
Green Tetris Block - L Shaped piece - Tetris
Pink Tetris Block - S Shaped piece - Tetris
Web with Eggs - Wall Creeper - TI
Strange-looking Mushroom - Anomura Fungus - TI
Teal Tetris Block - Reverse S Shaped piece - Tetris
Yellow Cloud in a Bottle - Flying nimbus - Dragon Ball
Yellow Tetris Block  - T shaped piece - Tetris