Where are the proof that Nintendo agree with them?
The problem with your logic is that you only count home consoles. Switch is both.
Switch is the new generation of 3DS and Wii U. 3DS is a 8gen console, you can't have Switch in the same generation as 3DS.
Nintendo doesn't agree. They don't say anything because they don't care, but that's not the same as agreement.
Wikipedia is edited by its users, so the gen 8 declaration on there merely reflects the consensus that the gaming community has reached. It's a consensus that was reached based on two big assumptions, namely that Nintendo consoles have short lifespans and that the PS4 and XB1 won't be replaced until 2021/2022. If those assumptions were correct, the PS5, Xbox 4 and Switch successor would all be launching in 2021/2022 and that's why Switch had to be generation 8 to preserve alignment.
From a Nintendo press release published at Business Wire:
"In September, the Nintendo Switch system was once again the top-selling current-generation console hardware, according to the NPD Group, which tracks video game sales in the United States."
And from Nintendo's very own website:
"Nintendo Switch has become the fastest-selling video game system of this hardware generation through 21 months, according to The NPD Group."
[emphasis added on both quotes]
There you have it. Straight from the horse's mouth. NPD classifies the Switch as current-gen alongside the PS4 & XBO. Nintendo have given their tacit approval to this classification. Ergo, the Switch is Gen 8. Game. Set. Match.
Re: Wikipedia. Nintendo press releases were used as a valid source for determining in which gen to classify the Switch.
Oh, and if you want to argue that Switch is Gen 9 because it comes after the Gen 8 Wii U, I understand the rationale. It's clean, simple, and intuitive. But reality is often messy, complicated, and non-intuitive, and that includes classifications (we see it all the time in biology). If we apply this "system's gen = predecessors gen +1" rationale to other systems, it becomes quickly ridiculous.
If we classify the Atari 2600 as Gen 2, then that means the 5200 is Gen 3 and the 7800 is Gen 4. But the 5200 was discontinued in May 1984, which was during the Big Crash, and over two years before the NES was released nationwide in the U.S. (and only 10 months after the Famicom was released in Japan, if you want to count that). The 7800 was released around the same time as the NES and Master System and was discontinued just over 16 months and four months, respectively, after the Genesis and SNES were released in NA. Do you really want to consider the 5200 being in the same gen as the NES and SMS, or the 7800 as being in the same gen as the SNES and Genesis?
Speaking of the Master System, unless you agree that the Sega Mark-III/SMS was an upgraded SG-1000, then would you claim that since the SG-1000 launched nearly concurrently with the Famicom that it is Gen 3 and that therefore the SMS is Gen 4 and the Genesis is Gen 5?
And that's not the only complication. To go back to Gen 2, long before everyone started referring to all pre-Crash of '83 cart-based systems as "Second Generation" (with Pong machines being Gen 1), some sources gave conflicting accounts of what system from back then belonged to which generation. Some have stated that the 2600, Odyssey², and Intellivision belonged to Gen 2, and that the ColecoVision and 5200 belonged to Gen 3, with the NES and SMS being Gen 4 (which would mean). But the 2600, despite being released in 1976, didn't really hit it big until around 1981 according to various sales figures I've seen, meaning it spent its best years competing directly with not only Intellivision but also the ColecoVision and even the 5200, which was intended to be a higher-end companion to the 2600 (the Intellivision and ColecoVision being more powerful systems), not necessarily a full replacement. Despite those older sources splitting the pre-Crash consoles into two distinct generations, the fact that we had all those systems being nominal competition against each other in the early 80s has led most of us to lump them together into a single "Gen 2."
It's clear that the NPD considers the Switch Gen 8 based on "competition" criteria. Nintendo had to ditch the Wii U prematurely because it failed, leading them to release a new system mid-gen. It has spent the last three years serving as nominal competition for the PS4 & XBO. By time the PS5 and XSX really start to take off, the Switch will likely already be well into the decline phase of its life. No Nintendo console has lasted more than about six years before being replaced (if you count handhelds, only the Game Boy lasted longer than that, but it was an anomaly and nobody expects a system to last a decade before being replaced). Assuming a Nov. 2023 release for its successor, that means it will have spent most of its primary lifespan, and nearly all of its best years, serving as nominal competition for the PS4 & XBO, not for the PS5 and XSX. If 2023 is indeed when the Switch's successor is released, then it will have launched closer to the launch of the PS5 and XSX than the Switch did to the launch of the PS4 & XBO. If Nintendo keeps running shorter gens, that will continue to complicate things. What if the PS5 releases in 2027 and the Switch's "grandchild" system releases in early 2029? Will we seriously argue that a Gen 11 system is coming out only 16 months after Sony releases their Gen 9 system?
If we're going to classify things, we need something that makes sense in the big picture, not what seems immediately simple and intuitive. Either we need to stop putting Nintendo systems into a specific generation, ditch the generation classification system entirely, or use some criteria that doesn't result in absurd situations over time or when applied to historical consoles.