OK, here you go, you knwo which genres I like (my examples from above), here are my 360 games:
Kameo
Viva Pinata 1-2
Banjo-Kazooie Nuts and Bolts
Sonic the Hedgehog
Sonic Unleashed
Sonic Generations
Mirrors Edge
Alice Madness Returns
Orange Box (may as well write Portal, I absolutely hate everything about fps)
Portal 2
both XBLA collections
Assassins Creed
Sonic & Sega Allstars Racing
Sega Mega Drive and Sega Dreamcast Collection
Child of Eden
Monkey Island Collection
(on Wii: Rayman Origins, you don't have to recommend that
)
Okay, so the common threads here are that you like visually bright and colourful games, you like Sega/Sonic, and you're not a fan of shooting or realistic driving.
Some recommendations--I've played all of these:
The Lego Series - The Lego games aren't particularly deep or complex, but they are funny, colourful, stress-free, and chock filled with content. My favourite is Lego Star Wars Complete, but there are many choices (3x LSW, 2x Lego Indiana Jones, Lego Pirates of the Caribbean, Lego Harry Potter, 2x Lego Batman). Disney Universe is sort of similar to the Lego games but not quite as good, so that's a secondary recommendation.
Adventures of Tintin - This is a side-scrolling 2d puzzle platformer sort of like the old Prince of Persia. I reviewed it more in depth
here just yesterday.
Bully - A T-rated open world game. It's like Grand Theft Auto Jr. What I like about Bully versus GTA is that there's less emphasis on combat and the world is a little smaller and more intimate. It's fun to ride around the town. The mission design is great. The story is fun.
Burnout Paradise - Burnout Paradise is a driving game, but it's not a realistic driving game. It's not a kart racer (if you want a kart racer, my recommendations would be Sonic which you already have, and then Blur, which you don't have) but it's very arcadey. It's fun, it's huge, and there's so much to do. Basically you have a big open world map like a Grand Theft Auto game, only you just drive around. Every street has an activity, whether it's beating the times others set on the street, or crashing through gates to side paths or jumping through stunt jumps, or doing races, or doing smash-em-ups. You can drive bikes or cars, and there are tons of different models. I don't like driving games either, but I liked Burnout Paradise.
Crackdown - Crackdown is an open world game with guns. But it's also got a cartoony kind of cel shaded style, and more importantly, it's as much a platformer as it is a shooter. Your character starts as a sort of super human with a huge jump and able to lift heavy objects, but as you progress you can practically fly and throw cars hundreds of feet into the air. The visuals are great, the world is huge and fun to play around in. It's just so fun to drive or jump off buildings or make explosions. Also, there are hundreds (800 in the first game, more in the second) of orbs hidden around the world waiting for you to find them. If you like collecting Jiggies in the platforming parts of Nuts and Bolts, you'll love this.
EA Create - Create isn't perfect and to be honest my girlfriend has played a lot more of it than I have. It's basically a sort of mix between LittleBigPlanet and The Incredible Machine. So it's a bright, colourful kind of puzzle / goldberg machine kind of game. You don't have a lot of puzzle games listed above, but on the off-chance you like light puzzle stuff, you'd probably get some use out of this.
Earth Defence Force 2017 - EDF is a shooter, but it's a very different kind of shooter than other games out there. It's very arcadey. You shoot huge masses of ugly looking bugs with tons of different guns. Your character repeats cheesy one-liners. It has an RPG kind of element where you grind armor drops and weapon drops. You don't need to be good at aiming. It's got split-screen too so great for local play.
Fable (series) - I'd recommend Fable II first. Fable is a very light action RPG with very simple combat. Besides combat, you can buy and manage property, marry and have kids, do quests, play minigames, etc. It's in a very colourful fantasy world. Colour-wise, I'd say Kameo is the closest comparison although Fable II also has some darker areas. It's cheap and I think the first part of the game is available as a free demo. Most of the complaints people have about the series is that they want deeper combat and more deep mechanics, but your taste here seems to tilt towards breezy games, and Fable II is excellent at being that. I had a really good time with it.
Harry Potter and the Order of the Phoenix / Harry Potter and the Half-Blood Prince - Colourful and easy introductions to open world games. Lots of content and fun stuff to do. Hanging around Hogwarts is kind of like hanging around the summer camp in Psychonauts. Good for a cheap price.
Kung Fu Panda - Light and breeze action / beat em up, great visuals and use of colour, good for a cheap price.
Majin and the Forbidden Kingdom - This is a nice sort of co-operative, puzzle based action-adventure game. It reminds me of a kind of alternate reality, low rent Zelda. It's got a lot of colour, it's very cute, it's pretty easy, and the puzzle to combat balance is very good. I think you can get it for less than $10.
Mini Ninjas - Action-adventure that has some stuff in common with action beat em ups and some stuff in common with Kameo. Nice use of colour. Not too hard, but pretty substantial in terms of content. Really cute.
Overlord (series) - Overlord is about half-way between something like Fable and something like Pikmin. If you liked Pikmin and you like humour, you'd like it. Very bright colours, very British feeling. It's got some light puzzle elements but it's mostly kind of an actiony game.
Portal 2 - If you like Portal, but want more of it, and a little bit less reflex based, with more story and more humour, Portal 2 is a great choice.
Prince of Persia 2008 - Imagine Assassin's Creed (1) only ten times more beautiful, with ten times less combat, and more of an emphasis on platforming and wall running in well designed levels rather than a big empty open world of buildings. It's not a perfect game, and some people hated it, but I loved it. It's so beautiful. If you like the movement in Assassin's Creed but not so much the game, please try this out. (I'd also recommend any of the three other Assassin's Creed games much more than the original, the step up in quality from 1 to 2 is amazing, if you didn't like AC1, you should still try AC2)
Rock Band - Okay, so, you don't have any rhythm games. Those are kind of one of the best thing about this generation. Maybe you don't like them. Maybe you don't have room. I don't know what your taste in music is. I don't know if you have friends over often. But if you want to have a good time with some people, learn some new music and get the chance to play some of the old music that you like, Rock Band is incredibly well made and a major player this generation.
Scene It? - Scene It is a series of three movie trivia games on Xbox 360. The third game is multiplatform, and is easily the worst of the three. You want the first or second one, both are excellent. If you like watching movies, and not just obscure foreign films, even very mainstream movies, Scene It has a great, colourful presentation. It's very fun to play with friends, but even solo is fun too. There are thousands of questions in each. The first and second games are available with a set of "big button" controllers, kind of like remote-style controllers, that make the games more accessible to people not familiar with the 360 controller.
Sherlock Holmes vs Jack the Ripper - So you don't have any adventure games listed, but I just wanted to put this here to highlight that the 360 has some genres you might not realize. Sherlock Holmes is a traditional, puzzle-based, inventory-based point and click adventure. The colours and story are quite dark (this is the tale of Jack the Ripper after all!) but it's visually different than just about everything on the system. I'm guessing you're in Europe because you say Mega Drive instead of Genesis, so
Grey Matter is also available over there.
Thrillville (series, although I think only Off The Rails is on 360) - Thrillville is a weird game. It's kind of like a theme park / rollercoaster building simulation, but it's also got another level--it's got a ton of minigames, some of which are very substantial, certainly moreso than Mario Party, but also it's got this level where you have to talk to people in your theme park with a pretty advanced dialogue system and kind of do sort of vaguely RPG-style quests with them. Very colourful, kid friendly, cheap.
Toy Story 3 - I must be nuts, recommending movie games. Toy Story 3 is probably the best kids movie game adaptation in a decade or more. The main game is your standard action platformer like every movie adaptation. It's over in 4 or 5 hours. It's better than average, but not great. BUUUUUT the real meat of the game is the 15-20+ hour town building simulator mode that's separate from the main game. You build a frontier town, populate it with people, do open world quests and missions, find tons of collectibles. I can't stress how much there is to do, it's amazing. It's not casual crap, it's a really, really good game.
The Burger King Games - I'm not sure if you can get these in Europe. There are three games: Pocketbike Racer, which is basically a kart-style racing game, Sneak King which is a stealth burger surprise puzzle simulator, and Big Bumpin which is an arcade-style bumper-car game. None of them are amazing, but they're all $5 or less. They're so surreal. I can't believe these exist. The last time a company made an advertising game like this, it was Pepsiman on PS1, which is also insane.
This is just to name 20 or so, and it's not an exhaustive list at all. Maybe you'll get through this list and say "I don't like any of it! Pwnd!" Everyone has different taste. But all of the games I've recommended to you here, I've played through personally, I enjoyed, and they definitely hit different niches than FPS or realistic racing.
I didn't list XBLA games that are only obtainable through download, but I also hate digital distribution, so I only buy download games when they are absolute must haves anyway.
To each their own, but realize that when you choose not to play certain games, it's going to negatively impact your impression of the platform. I'd recommend dozens and dozens of games here; Costume Quest, PB Winterbottom, Braid, Axel and Pixel, ilomilo, Adventures of Shuggy, A Kingdom for Keflings, Fruit Ninja, Stacking, Splosion Man, Polar Panic, Mercury HG, Cloning Clyde, Ancients of Ooga, Castlevania: SOTN, Torchlight, Lazy Raiders, Marble Blast Ultra, Worms, Outland, Raskulls, Limbo, The Maw, Scott Pilgrim, Roogoo, Carcassonne, Ticket to Ride, etc. To say nothing of Xbox Indie games, where I would also recommend dozens of titles.