Were there any third party games on Nintendo hardware that felt like "Nintendo" games to you even though they weren't developed or published by Nintendo, just cos they were so central and intrinsic to your experience of that console?

If so, what were they and what made them so special?

The Rare games that arrived a few years before or just after the Microsoft acquisition, for sure. Perfect Dark, Conker’s Bad Fur Day, Banjo Pilot, etc.

Also, the early Turok games — in part because they were Nintendo console exclusives and in part because of the connection between developer Iguana Entertainment and Retro Studios.

Outside of Rare, the Rogue Squadron games!

There have been a handful of HAL Laboratory games that they published exclusively for Nintendo hardware, but under their own banner. These included Adventures of Lolo, Day Dreamin' Davey, HAL's Hole in One Golf, Metal Slader Glory, and Trax.

Bayonetta 2 and 3.

Soul Calibur II

Final Fantasy Crystal Chronicles

The old classic rare games

Thats about it

Most games from another developer that come to mind was actually published by Nintendo.

But the Ace Attorney Games on Nintendo DS developed and published by Capcom. They felt like one of my main reasons for loving the system and the style felt Nintendo-ish. I hold them in as high regard as my most played games on the system, all of them published by Nintendo.

Plok (SNES - edit: looks like this may have been published by Nintendo in some regions)
Beetle Adventure Racing (N64)
Space Station Silicon Valley (N64)

The first two are exclusives, the last got a crappy PS1 port in Europe. For me, they're games I grew up with and are synonymous with the consoles in question. They all had the same kind of charm that you find in Nintendo games. Plok and Space Station also had that quirky nature that was similar to some of the Rare games were putting out on Nintendo consoles at the time.

Several Indie games come to mind, like Shantae, Shovel Knight or the Blob

Well some of the rare N64 games like Banjo, though Nintendo owned 49% of them back then.

Trying to think of a multi-plat and not sure there's a title I've personally played that fits the bill for me.

Modern examples I can think of would just be indies that are based on Nintendo titles. Like Wargroove being an Advance Wars clone.