Nintendo fans don't ask for much. They just want endless reiterations of the same core games and take what they can get from third party and that's it. It's a weird money printing machine.

Nintendo has games designed with mobile gaming in mind. No other big studio seems to be even slightly inclined to develop games with primary focus on handheld gaming

Crude way to put it, but you're not wrong with over 200 games in the Mario Franchise, 21 main series Zelda games plus also a lot of spin-offs and some 120 Pokemon games.

Endless reiterations with variations but keeping the core characters and gameplay philosophy the same. Familiarity is a great way to keep people, especially when keeping those core franchises exclusive.

They always know how to balance power and price with their handhelds very well. The competition mainly focused on power, and even in general, the most powerful console of the generation selling the best is a rare occurrence.

That and their software works really well on handhelds. Until the Switch there wasn't nothing that screams 'wow, I need to play this on my big TV', like Alex said. And with the Switch... well, you have that option.

As a grown ass man of 37 years that still love Nintendo games I think that one of the keys is the family focused games, or if you will, games marked mainly to kids.
It is the brand of choice to give to children. combine that with creating high quality products and you get a fan in that child for a long time.
Are you going to succeed over decades you probably need customers who buy your stuff for a large portion of their life.

A lot of it is luck. Wii or Switch could've failed, but they didn't. I mean any Nintendo platform could fail but those two were some big risks that paid off. Shoot, even Switch 2 has been a risk with mostly being an iterative successor. But risk was going to happen no matter what. Have Switch 2 iterative and you take risks, make it something radically different that might not even be Switch 2 and it's also a risk.
Handheld domination is a big part of it. From Game Boy onward, Nintendo has rarely been in any danger of failing in the handheld space or even being overtaken. Nintendo course corrected quite quickly when 3DS was slipping after launch. They gambled by selling it at a loss for a time and it paid off massively. 3DS helped keep Nintendo afloat, but with the failure of Wii U they needed to get into mobile and also spend a lot of time on the next system.
And it's the IPs as well. Look at all the Mario and Pokémon games. Core titles, tons of spinoffs, they print money.
Lower costs help too. For a lot of Nintendo's history (in both the handheld space and even recently in the home console space) they spend less on their games than most.

One Word best explains their continued success.

Exclusives

I think another aspect that has helped more times than hindered, is that they are willing to take a risk and instead of making yet another console that is just more powerful than the last, they have many times tried to use an outside of the box idea to offer new ways to play.

ie. motion controls, glasses free 3d, dual screens, "hybrid" console.

It's not just one thing.  It's a long of things which has made them a crapload of money:

1.  Strong both on the creative side and business side.
2.  Several IP with multiple strong entries each.  This means more than one generation will keep coming back to play these games.

3.  Total handheld domination, which was aided by focusing on gameplay over graphical power.
4.  The ability to make arcade style games well.

5.  The ability to innovate on hardware/controls and make compelling software that plays well with the new hardware/controls.
6.  Making broad games that can appeal to a variety of age groups.

7.  Making games that appeal to girls/women (e.g. Animal Crossing).
8.  Selling hardware at no loss or a tiny loss.  (I.e. rejecting the razor and blades model).

How you use the device is literally irrelevant.
My tablet hasn't left it's dock since I bought it... It doesn't change my tablet from being a mobile device.

Just how I use my device is different.

Nintendo's handhelds have always been high sellers backed by an amazing game library.

Difference between today and yesteryear is the competence of mobile hardware that allows for a decent image to be output to a larger canvas.

Good that we can agree.

The only rebuttal you can formulate against the Switch not being a handheld is one based on personal feelings.
If you don't see how that logic falls apart, then there really is no discussion to be had.

Nintendo's software is part and parcel with their hardware, both work together to make the platform what it is.

It's just a handheld with a USB-C dock... It's nothing special in that aspect, the dock doesn't do anything other than parse the image from the handheld to the TV, it doesn't add RAM or CPU/GPU resources to the console, it doesn't do any special upscaling or enhancements to the image... it's a "dumb device".

It's a plain USB-C dock for a mobile device.
No different than the dumb USB-C docks for my tablet, laptop or phone. It's all there is to it.

There are many that agree with me on this.

And there is also nothing wrong with it, it doesn't take anything away from the Switch... If anything it shows that Nintendo is adaptable to the market and realized a long time ago that it could never compete with Sony/Microsoft in the dedicated console marketplace, so they opted to innovate and be something different, something unique. 

And it's actually WHY Nintendo has been able to exist as a platform holder for so long, they are extremely flexible and adaptable to market conditions, they aren't afraid to take on risk.

That's a really pointless take in 2025.  The Switch is clearly designed as both a big screen TV console and a portable device.  There is simply no need to have different games and hardware for portable devices anymore. Just like PC games can run on both a desktop and a laptop: hardware limitations of the old days don't apply any more.