A major part you notice about the Nintendo Switch the more you use it, is that it borrows a lot of its DNA from older consoles, basically 6th gen and earlier. From its OS, to the games library, to the focus on local multiplayer, the Switch does feel very old school, yet new at the same time.
The first major sign of this, is its minimalist UI and OS functionality. Eschewing the multi-media entertainment and social hubs of modern consoles, the Switch presents an interface that can be described as a virtual game-shelf. You pick the game you want to play, watch a splash icon, and you're in. No omnipresent live-streams or social media feeds attached to these icons, just all your stuff laid out in a nice single file line. This extends to the features of the system as well. You have basic stuff like your friends list, a photo album for game screens, the eShop, and your standard settings. And that's it. For better or worse, there's none of the extensive list of features found on other devices, and while there are a couple media apps on the eShop, the Switch is a gaming device in the purest sense. It's the closest platform on the market today that replicates the feel of a retro console, while invoking modern elements as well.
And that also extends to the games. The Switch is not lacking in games, it's got over 1000 of them already. But it's limitations as a mobile device does mean it misses out on most of the more demanding AAA games found on other platforms. However, it has found comfort in other forms of software. Ports, Indie games, Mid-budget games, while these are available on other platforms, they really shine on Switch. The portable nature of the system lends itself well to these games. And things like 2D/3D Platformers, traditional RPGs, and puzzle games have a nice home on the system, as Switch versions of these types of games regularly outperform their PS4 and Xbox One counterparts. This is a case where the Switch's relative lack of power can work in its favor, as genres and game-play styles that have long since been abandoned by major AAA publishers, often stand out the most on Nintendo's system. This Switch has given a good home to some older classics from past systems as well with its remasters.
And then there's multiplayer. While you can play most Switch multiplayer games online, for better or worse, the service it offers is very rudimentary compared to the functionality players on other consoles are used to. No, where the Switch really shines is Local multiplayer. The system comes with Two detachable Joy-Con that can each be used as a self-contained mini-controller for multiple players. Combine that with the built-in Kickstand, and you have a multiplayer party machine, wherever you go. All the Nintendo multiplayer staples like Mario Kart and Smash are here, and third parties have found value in this feature with games like Fighters and Arcade games that you can easily play with a friend. Moreso than any previous console, the Switch actively encourages you to share your games with others locally, and it makes it very easy to do that.
These are the reasons I think, the Switch feels retro in spirit, while still having modern sensibilities.