Ok, I stand corrected.  However Sony was one of the founders of this association (and it should be obvious that Sony has advantages with the Blue Ray format).  So, there are subtleties to these points, but my points are still valid.

Since your Blu-ray argument is already debunked (and a Blu-ray drive probably ain't the first choice for a portable device anyways), feel free to give a few other examples of hardware components where Sony gets much better prices due to their "huge vertical monopoly" than Microsoft and Nintendo. Should be easy, since they own so much, that you could not even begin to tell us everything they own.

Special treatment is not really the advantage.  It's being able to heavily invest in the format from the get go.

Sony's greatest strength (compared to Nintendo) is their global presence. They have over 100K employees across the world, lot's of partnerships with (local) distributors, factories and what not. For Sony it's much easier to penetrate the markets of developing countries like East-Europe, Middle East, Russia, Latin-America, etc. 

Nintendo really is a very small company when you come to think of it. They only have around 5K employees, which is less than Ubisoft, EA, Activision and the same size as Sega. And these companies don't even have a hardware department like Nintendo does. 

Microsoft's biggest weakness is their lack of presence in Japan and lack of partnerships with Japanese developers. That means they lose out on a ton of high quality games like Dragon Quest XI, Persona 5, Nier Automata, Yakuza, Bayonetta, many jrpgs, etc.

These might not be the biggest franchises, but all these franchises together can make quite the difference.

According to this estimate the most costly components of the PS4 were the CPU, GPU, RAM and hard drive:

A hybrid would add a display and battery.

For which of these components Sony has a "vertical monopoly"?

They aren't in the GPU, CPU, RAM or Hard drive business. They are in the memory card business, but when we look to the Vita and PSP (Memory Stick Duo), their memory solutions were never competitive to SD and MicroSD.

Their own manufacturing plants for mobile displays were shut down years ago: https://en.wikipedia.org/wiki/Sony_Mobile_Display 

The Vita displays and most Xperia displays are made by Samsung instead of Japan Display, Inc.

One of the very few Sony components in the Vita and the Xperia phones is the battery.

a) Battery is a non-trivial cost for a portable PS4.  The Switch already eats through its batter far faster than any previous Nintendo handheld.

b) There is no way I am going to go through all of Sony's holdings to answer these questions, because they own a crap ton of stuff.  If looking at this stuff is the sort of thing that interests you, then be my guest.
https://en.wikipedia.org/wiki/List_of_acquisitions_by_Sony_Corporation

Again, it does run fine on 360. It has frame dips but it also achieves fairly stable 30 fps in less stressed areas. The frame rate doesent dive like GTAIV. DF said the PS3 is the overall superior version, they didnt say the 360 version runs poorly like you suggest. Hence, it seems run well in on 360 in spite of being built for PS3.

Again, Im willing to consider LA Noire is a poorly optimized game but thats one of few existing ports. It demonstrastes the superior GPU, but its not really 1080p and struggles to maintain 30 fps even with more power.

Again, BF4 may have lower quality effects but it also has more impressive assets, AI, large enviirnments, resolution, etc. It actually looks more polished than Doom on Switch.

The X1 games I mentioned varied from 900p-1080p. The list of 720p games is actually pretty small, especially thanks to dynamic resolutions. Its also worth noting that MS eventually unlocked resources dedicated to Kinect and X1 specs are also kinda unique due to ESRAM or something like that. Either way, X1 had impressive games in its first year which you seem to have forgot.

I'd argue older games like Killzone SF and Infamous SS are comprable in graphics to Uncharted 4 and Horizon ZD. I mean these are all great looking 8th gen games.

In practice, I still see Switch capabilities closer to 7th gen than 8th gen. I'm curious to see how it handles more 8th gen content because Doom was functional but showed significant limitations.

Sorry, but the battery is not a big cost factor in a mobile device. The hybrid would probably have the capacity of an iPad (or less), which costs $10 - $15... perhaps Sony could save $5 from that. Bigger batteries would make the hybrid too heavy for handheld usage.

PS3 and 360 version of games have unstable and frame rate problems, we saw that in DF video, so that's a fact, so no need to talk about that anymore. I didnt said runs poorly. Again, PS3/360 definatly had better optimisation, like I wrote before Rockstar released LA Noire on PS3/360 they already were working several years on PS3/360 hardware and released couple games already, also they have similar architecture, while this is of their 1st year Switch game and totally different architecture.

So what if its not locked 1080p, dynamic 1440x1080-1920x1080 resolution compared just to 720p make huge difrence, also dont forget better textures, shadows and ambient occlusion.

Key word here looks less polished, and I agree, I think that Doom port on Switch did not had best optimisation, while BF4 definitely had great optimisation on PS3/360. Here is again point about 1st year ports compared to 5. year ports.

You are wrong, look at list of XB1 games from 1st year and you will see that most of them runs at 720p-900p not at 1080p, I will just mentione some of 1st year XB1 720p games: BF4, CoD Ghotst, Dead Rising 3,  Watch Dogs, Killer Instinct, Metal Gear Solid: Ground Zeroes.

Uncharted 4 and Horizon ZD looks much better than any other PS4 game, and thats not strange, beacuse in later years we always geting more demadining and better looking games than we had in 1st year of hardware on market.

All we saw until now shows us that Switch in capability is somewhere between PS3/360 and XB1, and with evre new game (more advanced, better looking, more optimised games) difrence can be only bigger compared to PS3/360 games.

Again, I said LA Noire demonstrates the superior CPU of Switch with higher resolutions and effects. But there is also frame drops, slow down, and I believe lowered draw distance. Whatever, its fine on Switch but it also shows potential limitations of the hardware.

I mentioned the resolution because its pushed as a 1080p game by some, its not.

BF4 is older and on inferior specs. But it looks more polished than Doom on Switch. Thats all I'm pointing out. Not really even making arguments, perhaps there are valid reasons, just making that point. Doom on Switch reminds me Destiny on 7th gen. Its technically impressive, has advanced lighting we rarely saw on the specs, even a rock solid 30 fps, but lacks polish and sub HD.

Thats not most of X1's first year library, thats a few. Four of which are 60 fps. The X1 doing 60 fps with modern graphics tends to be around 720p-1080p. All of the 60 fps games could have been 1080p at 30 fps. DR3 and Watchdogs are poorly optimized, I didnt mention them for a reason. I mentioned impressive 1st year games.

Eh... Uncharted 4 and Horizon ZD look great. But its not a generation leap or anything. They look like a polished 8th gen games.

Right now... I think we still need to see more ports on Switch. Especially 8th gen to Switch, not just 7th gen polish.