Can I use arduino due and it's genuino versions as a gamepad/mouse etc.?

I have an arduino leonardo and uno. I am thinking about a big project, but I thought an analog pin extender isnt worth it, since I couldn't even see a good tutorial on them.

I did notice that the arduino due had the same port as the leonardo and pro micro.

That made me wonder: will the arduino due work as an HID device?

Yes, you can do that. Just be sure to connect the "Native USB Port" to the host device when you want to do HID. You can't do it over the "Programming Port". On the Leonardo and Pro Micro there is only the one port, so there's no chance of confusion with those boards.

So every genuino variant of an arduino board is almost the same, with the difference being colours and branding? A genuino due/pro micro should also work like this?

Absolutely not. The different Arduino boards provide a huge range of capabilities, accessories, form factors, and prices. It happens that both the Due and Pro Micro have HID support, but there are also major differences between those boards.

You can find more information at the product pages for the various boards in the Arduino Store:
https://store.arduino.cc/
Make sure to also check the board's "Getting Started" page, which is linked from the store page.

FYI, "Genuino" was a trademark the Arduino company established as a worst case scenario fallback during a legal dispute over the "Arduino" trademark. That legal dispute was resolved years ago and the "Arduino" trademark is here to stay, so the "Genuino" trademark is only relevant for historical purposes and is not something that makes sense to use in the context of our current discussion.

Maybe I misunderstood. Are you asking whether there is any difference between an "Arduino" branded Due and a "Genuino" branded Due? If so, you're right that it's only a matter of branding.

Some Arduino boards have gone through revisions over the years (usually very minor things that are not really of interest to the users). Since Genuino branded boards are no longer being manufactured (though some of Arduino's distributors might still have inventory of those boards), there might be some correlation between a revision and a brand, but this would be purely coincidental rather than some intentional association between the trademark and revision.

I actually thought genuino was the name of arduino compatible boards that weren't made by arduino. I wanted to know if boards that look like the due, and pro micro (the exact same port placement, shape) with the only difference being colours, price and branding have the same capabilities.

For example,

This board is about 4x times cheaper than arduino due and I wanted to know if I could use it for HID and other things the due is capable of.

Ah, OK I see now. There are four classifications for unofficial boards:

  • Compatible
  • Derivative
  • Clone
  • Counterfeit

You can learn more about these in this article:

In the case of true clones, this is correct. By definition, a clone is physically identical to the board it clones, with only cosmetic differences. The board you linked to appears to be a clone. However, derivative boards are very often referred to as "clones", even though it is not accurate to call them clones.

In the case of derivative boards, they may look similar to an official board, but by definition a derivative has some meaningful difference even though it is based on another board.