The words "overlap", "primarily" and "includes" all lead to Nintendo being direct competitors to Sony and Microsoft.

The issue that many people seem to be having is they think "direct competitor" is the same as "carbon copy".  In Generation 5, Nintendo, Sony and Sega were all direct competitors but none of them tried to be carbon copies with one another.  The N64 and PS1 had fairly different game libraries in a lot of ways, but they were directly competing with one another, because they were both targeting the same types of customers with a dedicated home system.  Then Microsoft comes along as the carbon copy of Sony and also Sega drops out of the hardware business.  All of the sudden Nintendo looks really different, because Microsoft and Sony are so similar.  The reality is that Nintendo and Sony didn't change.  They are still directly competing.  But Microsoft being a carbon copy of Sony now makes Nintendo look more different.

It is very much like this.  McDonalds, Taco Bell and KFC all directly compete in the same town, because they are all trying to sell fast, cheap, lower quality food.  If KFC goes away and Burger King comes along, then that doesn't change that McDonald's and Taco Bell are still directly competing.  Sure McDonald's and Burger King look really similar, but Taco Bell is still selling to people who want fast, cheap, lower quality food.  Olive Garden is actually an indirect competitor.  It's still a restaurant, but it's slow, more expensive and higher quality food.  The type of food (burgers or tacos) is actually not as important as price, convenience and quality.

Nintendo, Sony and Microsoft all make dedicated home systems.  They are all of similar price, convenience and quality.  They compete directly.  Meanwhile smartphones and PC's are indirect competitors.  Smartphone gaming is cheaper, more convenient and is a lower quality experience compared to consoles.  PC's on the other hand tend to be less convenient than a console, but can give a higher quality performance.  PC's can even have really similar games to consoles, but they've always been in different markets.  What matters more is price, convenience and quality.

Answer me: Are fans of Killzone/God of War/Halo/Gears of War/Fifa/CoD/GTA/Battlefield and fans of Mario/Zelda/Animal Crossing/Fire Emblem/Pokemon the same? 

They have overlap but they are aimed at different demographics.

PS/XB are aimed primarily at the 13-35 year old male demographic that plays online military shooters, sports/racing sims & open-world/cinematic action games.

Nintendo aims for a more broad demographic which includes children, females & families with things like platformers, turn based games, arcade style games, local multiplayer games, puzzle games, fitness/rhythm games and social sims.

So while they compete in the general sense that they are video game devices aimed at video game players, they are not direct competitors because they do not compete for the same demographics.

You need to learn the meaning of competition. It doesn't matter if the console's libraries are completely exclusive, completely multiplatform or any % in between. We have similar products competing for the consumer's money. Most people buy a console INSTEAD of another and then again the following generation and so on.

The problem with this argument is that all 3 are actually competing for all the demographics.  When Nintendo has exclusives like Bayonnetta or Xenoblade Chronicles, they are competing for the 13-35 year old male.  Not to mention that this category really likes games like BotW or Smash Bros as well.  And when Sony or Microsoft produces games like Little Big Planet, Knack or anything with the Minecraft label, then they are competing outside of the 13-35 male demographic.  On top of this all 3 have plenty of third party games that appeal to a variety of demographics.  

All 3 companies are trying to get all of the current gaming market onto their platform.  They might each have a different focus, but they are similar enough to still be directly competing.  You might as well say Netflix, Hulu, Apple+ and Disney+ are not competing streaming platforms.  They each have a different focus, but they still directly compete with one another.  They are offering a similar type of product and aiming it at the same broad group of consumers.  These companies are directly competing.  The big 3 gaming companies are doing the same thing.

Look at the top seller games of Switch and PS4/Xbox 1 again. If they are the same, then i agree that they are direct competitor. Switch top seller is very different than ps4/xbox 1 top seller. People who have interest in Cod, Fifa and GTA won't get Switch to play Mario, Pokemon or Fire Emblem.  

At the bold: just like Wii - Wii U and Xbox 360 - Xbox One, even PS2 - PS3, agree

This is why i already said they are direct competitors in some way of software and that's about it. BOTW vs Horizon is an example. But Smash? Where is PS All Star on ps4? Bayonetta and XC are not top selling of Switch games, neither LBP nor Knack on ps4. The top selling games on Switch (1st party) have nothing to do with the top selling games on ps4 (3rd party)

You need to learn the meaning of competition. It doesn't matter if the console's libraries are completely exclusive, completely multiplatform or any % in between. We have similar products competing for the consumer's money. Most people buy a console INSTEAD of another and then again the following generation and so on.

The problem with this argument is that all 3 are actually competing for all the demographics.  When Nintendo has exclusives like Bayonnetta or Xenoblade Chronicles, they are competing for the 13-35 year old male.  Not to mention that this category really likes games like BotW or Smash Bros as well.  And when Sony or Microsoft produces games like Little Big Planet, Knack or anything with the Minecraft label, then they are competing outside of the 13-35 male demographic.  On top of this all 3 have plenty of third party games that appeal to a variety of demographics.  

All 3 companies are trying to get all of the current gaming market onto their platform.  They might each have a different focus, but they are similar enough to still be directly competing.  You might as well say Netflix, Hulu, Apple+ and Disney+ are not competing streaming platforms.  They each have a different focus, but they still directly compete with one another.  They are offering a similar type of product and aiming it at the same broad group of consumers.  These companies are directly competing.  The big 3 gaming companies are doing the same thing.

Even though both BOTW and HZD are great games I would that even they don't compete much in sales and that most of the buyers of one weren't looking to buy the other and decided not because of it.

I think they are in some way because both are action open world, both are some of biggest games in 2017, both released in early 2017

Look at the top seller games of Switch and PS4/Xbox 1 again. If they are the same, then i agree that they are direct competitor. Switch top seller is very different than ps4/xbox 1 top seller. People who have interest in Cod, Fifa and GTA won't get Switch to play Mario, Pokemon or Fire Emblem.  

At the bold: just like Wii - Wii U and Xbox 360 - Xbox One, even PS2 - PS3, agree

This is why i already said they are direct competitors in some way of software and that's about it. BOTW vs Horizon is an example. But Smash? Where is PS All Star on ps4? Bayonetta and XC are not top selling of Switch games, neither LBP nor Knack on ps4. The top selling games on Switch (1st party) have nothing to do with the top selling games on ps4 (3rd party)

Sure I can conceed they compete in some way, being similar genre/idea/gameplay (sure plenty differences exist). But I don't think a significant enough amount of people decided not to buy BOTW or HZD because of the other (BOTW broke all records for Zelda and HZD also broke for new IP 1st party for Sony at the time and also sold much higher than any previous GG game).