Well, Switch cannot be Generation 8 (PS4, XBoxOne), simply because it was released so much later, on March 2017 if I'm not wrong, it cannot be the same generation of PS4 and X1 by "definition". Generation 9 ? nope, 'cause PS5 and XBox Scarlet family will most probably be released on Nov 2020, 3 years and half after Switch. So which Gen is Nintendo Switch ? Switch fits in its own generation, with its own game experience and features, and it does not compete directly with Sony or MS, and also all the comparison in sales among them make no sense, as Sony and MS are getting ready for Next Gen, while Switch is a new console.
I would define the Gen of Switch as an hybrid Gen.

So how do I play my WiiU games on Switch or my Switch games on WiiU? Where is the compatibility?

Or we can say we are at Hybrid gen1 and VR gen 1 ;)

I remember that at the time it was called the 128bit "next gen".

Maybe you are right about the cpu specs, they are not 128bit, but overall the architecture were 128bit or at least the compute unit in the GPUs. 

Dreamcast "128-bit graphics-oriented floating-point unit delivering 1.4 GFLOPS" wiki

Gamecube "The Dolphin platform is reputed to be king of the hill in terms of graphics and video performance with 128-bit architecture." wiki 

Playstation2 "Communications between the MIPS core, the two VPUs, GIF, memory controller and other units is handled by a 128-bit wide internal data bus running at half the clock frequency of the Emotion Engine but, to offer greater bandwidth, there is also a 128-bit dedicated path between the CPU and VPU0 and a 128-bit dedicated path between VPU1 and GIF. At 150 MHz, the internal data bus provides a maximum theoretical bandwidth of 2.4 GB/s. Vector processing unit registers: 128-bit wide, 32 entries" wiki 

It was marketed as 128 bit gen. XD

And that is why they are specialists and us mere users =p

So how many generations must we add on the Handhelds due to Game Boys versions that older GBs couldn't play some games from newer, DS1 and N3DS?

Game&Watch -1

GB -2

Virtual Boy -3

GBC - 4

GBA - 5

DS - 6

DSi - 7

3DS - 8

N3DS - 9

Switch - 10

Is that how you would classify gen for Nintendo Handhelds?

Plus it seems like you just ignored the part of Virtua Boy and WiiU being killed of prematurely and being replaced, plus the HW inside Switch being in line with what we have in gen 8 but won't be for gen 9.

Switch is like a relaunch of their 8 Generation console, so it's Gen 8.

Honestly, in retrospect, WiiU almost didn't exist. Even Nintendo is treating it as such. It is like a prolonged Virtual Boy period. WiiU is like a Switch Beta period, and I think game history will as well in another 20 years.

First, I think it is more Playstation fans than Nintendo fans claiming that it is generation 9.  Secondly, changing the generation wouldn't change the fact that it will be outdated by PS5/XBoxTwo.  Everyone is aware that the Switch is barely performing in the same class power-wise as current gen systems.  Putting it is generation 8 or 9 won't change that fact.  It also won't dictate when the switch successor is released.  Sales and game development will.

So which gen was the Vita in your opinion? It was already outdated by the Xbox 360 and PS3 from the start and there is clearly a generational leap between Vita and Switch (both mobile hardware with battery constraints, volume constraints and thermal constraints).

Virtual boy didn't have any mainstream games, all its games were focused on a special gimmick/feature... it is similar to EyeToy, PS Move, Kinect and PSVR, which all are additions to the "normal" games. It also wasn't very mobile/portable so I'm irritated that you count it to the handheld generations.

Over 95% of the DSi game library also ran on the DS, over 95% of the new3DS library also ran on the 3DS.

0% of the WiiU or 3DS games run on the Switch. 0% of the Switch games run on the 3DS/n3DS or WiiU.

Do you really want to claim that these situations are the same?