Do I have to? It's pretty obvious...
1 - Nintendo is the oldest player in this game. You never subestimate Nintendo.
2 - On the more tangible reasons, any company that is inside the gaming industry is competing with each other, dierctly and indirectly. And that's especially true for MS, Sony and Nintendo, which are all console manufacturers. MS and Sony are now catching up with Nintendo, entering into the 9th gen, which Nintendo has a "3 year advantage". Not only Nintendo has a bigger library that uses the Switch full capabilities, but it's an extremely strong library of first and third party games.Not only that, but it's biggest gun going into this holiday it's that it's going to be much cheaper than the PS5 and the Series X, potentially being 250 dollars during Black Friday.
When Nintendo was weak(During Wii U), it not only beat Sony in the handheld market, but it came back strong with the Switch. After countless failures, Nintendo never left the console war, while MS seems to be going the Sega route(which would eliminate it as a competitor, as far as Sony is concerned). And the the most important part of the industry, the software, is where Nintendo is easily the strongest of the three, in sales numbers, quality, quantity and variety.
Sony and MS may be similar in the hardware design, but that alone dosen't mean that they are their only direct competitors.They are first and foremost a hardware maker, and thus it makes Nintendo a direct competitor too. Add to that Nintendo strenghts, and they are easily it's strongest foe.
You said that Nintendo's the true biggest competitor. This is not a very common perception and not pretty obvious. When the console market is analysed, even by professionals, is always in terms of Xbox vs PS.
Nintendo is getting far different than Sony and MS since GC. With Switch, what Nintendo is selling is totally different than current gen consoles.
If we consider every platform that sell games a competitor. Yes, Nintendo, Google, Apple, Steam, etc are in the game. However, in terms of current generation classic console with the standard 3rd party library, there are only two choices: Playstation and Xbox.
It's obvious to see that Playstation owners might also buy Nintendo, Apple, Android, Google, Steam games. The same for Xbox owners. But Xbox owners usually don't buy PS and vice versa. So they're directing competing for the same audience. That's how they're each other's biggest competitors. Nintendo is a competitor, but not really the biggest competitor.
In business the perception is very rarely the reality. I remember watching a talk by Peter Thiel (Paypal cofounder) where he said every business in a highly competitive field is pretending to be a monopoly. Meanwhile every monopoly is pretending to be in a highly competitive field. One example he gave was Google, which has a monopoly on search engines, so they portray themselves as an advertising platform. So many businesses are trying to misrepresent the reality, and sometimes the other "players" like analysts and journalists are actually doubling as marketers.
So the reality for gaming is that Nintendo and Sony are the two biggest players and both of these companies realize it even if they don't let on. Microsoft is a significant player, but they've never had success in gaming anywhere near the other two. How many generations has Microsoft won? Zero. Have they ever had a handheld system? Nope. They are, at best, as significant as Sega in their prime, but even Sega had a handheld system.
Meanwhile both Nintendo and Sony have a track record worth talking about. Nintendo has won 50% of home generations that they were in (NES, SNES, Wii). They've won 100% of handheld generations. That's a very impressive track record. Sony has won 75% of home generations that they were in (PS1, PS2, PS4), but they've won 0% of handheld generations. Also, Sony successfully defeated Sega permanently in the home console market and that is like double points when you defeat a competitor for good. However, Nintendo defeated Sony permanently in the handheld console market and that is again, like a double win. Nintendo is the most dominant player in the video game industry, but Sony is a very worthy competitor.
Meanwhile, the perception is that Sony and Microsoft are the big players and Nintendo is this quirky little company doing it's own thing. Nothing is further from the truth. Microsoft is actually the one trying the weird, quirky strategies because they are trying to move all of gaming online. They tried doing it at XB1 launch and they are trying it again with Gamepass. Microsoft is very good about controlling the messaging on the internet, but they aren't so hot at actually understanding gaming, so I expect Gamepass to be another failed experiment. However, Nintendo permanently defeated Sony in the handheld market, and then they launched the Switch. The Switch, being a hybrid, means they can make a handheld device that also invades the home market. That is the real threat to Sony. It isn't Gamepass. It's the Switch.
People think that Nintendo hasn't been competing with Sony and Microsoft since the Wii, but that begs a question. "Why did Sony and Microsoft make motion control devices?" It's because they considered themselves to be directly competing with Nintendo, even in the Wii era. Nintendo doesn't make and market keyboard and mouse controls to compete with Steam, yet Sony and Microsoft did make and market motion controls to compete with Nintendo. These 3 console makers all consider themselves competitors, but they also try to portray something very different to the rest of us.