I don’t see Nintendo/Game Freak doing something about Palword. Due to Palworld's first trailer release two years ago. So if Nintendo/Game Freak was going to do something about Palworld. They had two years to do it.
They can try, nintendo never did, doubt they will win, and cost them the decades of good will they've gained. The genre name is either a part of the title mixed with the word clone, like, lite, or a portmanteau like Metroidvania, so it's not 100% case.
That's why Palworld did not mentioned Pokemon on the steam page and the trailer.
It's also good for business. If someone comes in not knowing about Souls games and starts seeing it, they'll wonder why it's called that.
I played both Metroid and Castlevania when I was a kid so I knew where those came from. I quit gaming for a decade or so and didn't know anything about Souls-like games when I got back. I started seeing it everywhere and it made me curious. Now I'm a fan of From and games like what they put out.
Can you imagine the ego stroking it must be to have an entire genre named after your work?
They probably could have had a case against the first game to market itself as such, but I think once the terminology gets utilized as a "common term", you can't do anything. It's why Nintendo in the 80s and 90s was so fucking adamant on pushing parents away from calling any and every home game console a Nintendo
Fromsoft would have 0 ground if they wanted to today, but honestly, again, I think it's one of those things thats such a pedestal that there would be no reason to go after it.
There's also the matter that they aped BOTWs sound library in a way that crosses from "inspired" to "plagiarism", but that's easier to fix.
The argument for IP stealing wouldn't be so much the designs alone, but the intention. The "Pokemon with Guns" thing might be used as evidence that they were deliberately copying Pokemon's designs and tweaking them a bit. That's the same difference ad between reading the same source materials and copying your homework while replacing a few words.
INAL and it's been ages i studied copyright law and it's also depend on country(in this case, it's japan company vs japan company), but plagiarism only catch anything on the surface, they have no reach on the concept or idea or thought itself. The work itself also have to be very, very close to the original artwork, otherwise it can considered derivative and is protected by copyright law. In this case, Palworld have their own idea and twist on their monster even though it looks like the original work they copied, and the gameplay are mostly unfamiliar from the original work. They make it close enough to evoke the same feeling people have toward pokemon, but different enough to get away with it.
This is why Nintendo did not try anything against Genshin, even though Genshin took BOTW and add some of their idea. And this is also why you can film your own version of lion king, but the character is all bugs or something.
Personally i think Pokemon Company will not take action, the statement could just be something to make the fans stop bothering them. But i'd like to proven wrong.