The thing is, what IS "anime style?"
The industry varies very widely in styles, from Bebop, to Berserk, to Sailor Moon, to K-On, to Lucky Star, to Monster, to Claymore, to Prison School, to One Piece, to Dragonball, each and every one of those varies very drastically from the other, and there is no one way to pin it exactly. Then you have Western Cartoons which have been animated by Japanese studios, like Swat Kats, to Transformers, to Teen Titans, to the Powerpuff girls, and likewise, you then have western shows that take up Oriental influences like The Last Airbender and Samurai Jack, and the lines between "anime" and "not anime" begin to really blur.
Moreover, you have to keep in mind that Japanese animation began by copying French animation techniques, and then Osamu Tezuka copied the style of Disney with it's characters featuring large eyes to create Astro boy, and now modern Disney copying styles from Kyoani, and it all becomes a jumbled mess.
The west has always influence the east since its conception, and the east, having the largest industry, has always animated for the west, and moreover, influenced it back.
So, keeping all that in mind.
What is "anime style?"
Saying "Zelda has always been influenced by anime style" because the look of the series has donned the look of Japanese artists doesn't mean it's always had an "anime style." Artists in the East tend to influence each other a lot, so all modern oriental art styles tend to share similarities. Whether they be for animation, or vidya. This doesn't mean that they're purpsely trying to do an "anime style", it's just the way people draw over there because of their artistic influences.
Compare the art of Twilight Princess with its more realistic look and proportions, and detailed character designs, it could be compared to Moria's Berserk, which itself has very strong western influences in style. Likewise with WW as people have brought up, the hyper-deformed cartoon style would be more welcome in the west whose animation industry tends to prefer super deformed and simplistic designs due to lower budgets and tighter schedules, see:
Of course Japanese studios are gonna be influenced by Japanese artists, that's what they're around all the time. However, it should be noted that in this specific instance, Aonuma has said that they specifically went out of their way to seek a more "anime-like style." What he means by that exactly? Beats me. But I presume he's not referring to Japanese animation exclusively, but rather all animation in general due to their use of cell shading, which is very common in games trying to port the 2D look of animation onto 3D models.