If Nintendo and Sony had only worked on that Nintendo Station...there'd only be need for one console.

You're statement is kind of a contradiction though. Look at Sega of today ... they're a shell of a company. 

There was too much competetion for them to stay in the business. 

Over competetion is actually not a good thing, what it means in the game business is a very good company is going to get pushed out no matter what they do. People will only buy a certain amount of consoles and handhelds per generation. 

Nintendo needs to protect themselves and stay ahead of the curve, while they're not nearly as poorly managed as Sega, if the handheld erosion does not stop (and I'm not sure it will, if kids are choosing a tablet over a 3DS at age 7, I don't think they'll magically switch at age 9/10/11/12) they could find themselves in a pretty bad spot sooner than later. 

More competetion is better IMO is actually one of those statements in the game business that's actually false. If Nintendo is basically forced to dismantle itself because of over competetion, to me that's not good for the gamer. And for what? Microsoft and Apple maybe get a little richer but they don't need the game business anyway. 

Something like this will probably never happen, but it's  really interesting to think about it!

Had sega stuck with dreamcast maybe they wouldnt be in the situation they are now we need 3 companies plus nintendo games suit nintendo controllers.

I think we already know that Sony IPs probably would have sold better on 3DS than they did the Vita.

If Sony did that with N3DS or 3DS successor, it could happen - but Nintendo won't just choose Sony to partner for home consoles, there need to be other serious incentives on the line

Sega could've done everything right with the Dreamcast, they poured their heart and soul into that console, and people still wouldn't have bought it because they were hellbent on getting a PS2. That's all there was to it. 

A console is not like a soda brand where one day you can try Pepsi and the next day maybe you have a Coke or Sprite. 

People buy generally one console and that's their console for 5 years. A minority buy two consoles. Very few have more than that. 

In said scenario a perfectly good company can be forced out of the industry. You only need two good competitors to have healthy competetion in the market, three-four is unsustainble over time IMO. 

"Franchises like Mario/Mario Kart/Zelda deserve to be available to more people"

They're already available to anyone who wants them.

Making a lot more money is a pretty big incentive. Amiibos would sell a hell of a lot more under this setup. So would Nintendo software like Mario Kart and Splatoon and Zelda. Having one less competitor is another nice incentive. 

Potentially saving the Japanese game market also is a strong incentive for Nintendo. There's many, more than I think people realize, it's not just "oh well, we got Uncharted on our handheld now", that's more of just a bonus. 

There are lots of other ancilliary bonuses too ... Sony is a great parts manufacturer (Apple uses them for their iPhone cameras), Nintendo could use them for LCDs, cameras, battery, etc. They get a good price, Sony gets a good vendor for their parts, win-win. Sony loves to market their partner exclusives or even when they're not exclusive, they did a great job with Destiny. I imagine they'd be keen to help market things like Zelda or Mario Kart just as much. 

There are lots of incentives, more than I could list in the OP. 

I don't think so. These are mainstream IP that everyone knows and many people have grown up on. 

Nintendo's consoles are the problem, not their IP. People love the IP. They just don't want to have to buy a seperate, gimmick-ey $300 console to play them on.

Put a few of these bigger IP on the PS4 and let Sony co-market them like they did with Destiny and you would see huge sales. I have zero doubt about that. 

Yeah but like soda sometimes u feel like a coke or go for sprite one day or feel like a milkshake choice is good even if u only have one console least u have a choice we dont want a dictatorship.

Also Nintendo is kids games Sony boss yosp said this we need nintendo but for the love of god there next console needs a sensible controller and graphics near the others consoles.