It is well within the Pokemon Company's legal right to have Pokemon Showdown taken down. The simulator makes unlawful use of their copyrighted assets in the same way Pixelmon and Brick Bronze did, and is technically illegal. However, whether it will actually have action taken against it is a very different question.
At this point, Nintendo and co would all be aware that the simulator exists, and is as popular as it is. They could drop the hammer at any moment. However, there is a multitude of reasons why this may not be in their interest, or that they should at least play the waiting game for some time. Some of these include:
Showdown being removed would get a massive press run. Nintendo is already notorious for their stance against derivative fan projects, but Showdown would take things to a new level. Nintendo might not want the removal of such a massive project behind their name.
Showdown is basically free marketing. While it does compete with the real games on some level (and infringes on their right to create a similar product with their property), Showdown's accessibility and spread across the internet may be more valuable than whatever they lose in sales.
Removing Showdown would be a massive dent to the competitive community. Nothing of Showdown's size or quality exists in official Pokemon media. If they're interested in building Pokemon as a competitive game, getting rid of Showdown would appear counter-intuitive.
Even if it was removed, Showdown would be replaced instantly. Everyone would move to the next biggest simulator, and they'd have the same problem again. Removing multiple simulators would only make the other issues worse, and would be a bother for them to deal with as a whole.
tl;dr it's illegal, but they probably won't do anything about it.