As @HGohwell said, Japanese media is always being influenced by overseas media and vice-versa. This has been going on for thousands of years, and was actually the subject of my English Literature dissertation in university, where I described it as a "cultural tennis match". 
Japan, like the UK, is an island nation just off the coast of a continent with a bunch of larger nations, and so has a preoccupation with its own uniqueness and identity. This tends to lend itself to a whimsical self-expression that can be charming, or can be obnoxious (especially when the whimsy and charm is juxtaposed against a past of imperialist expansion and oppression, another trait we English share with the Japanese...).
One of the most distinctive things about Japanese culture that I found when living and studying there is something the exchange students started calling "Existing in the moment of doing." Japanese society and art is all about appreciation of the process of doing a thing. It's not that they don't care about the outcome; it's that they believe that if you learn to do the process perfectly, master doing every step of it properly, a good outcome is a natural conclusion.
This approach has both strengths and weaknesses. A strength is that Japanese media has a careful attention to small details, and can effectively use and appreciate quiet moments that a Western creator or studio might cut because they "don't add anything to the finished product". A weakness is an overreliance on tropes and archetypes or stock phrases and plot beats or specific stylistic shorthand that can make a lot of lower-quality anime and manga really start to blend together.
It's a tricky thing. I love Japanese media, and when I first got into it as a teenager in the late 90s, it really was mind-blowingly fresh and unique. As I've got older though, I've got more and more picky, because once the tropes become familiar, so much of it really does become a bland mass of similar stories with similar archetypes. I feel like also the anime industry has stagnated somewhat, and become a bit insular and weird... too focused on making stuff to appeal to otaku rather than making stuff to tell a sincere and meaningful story. I think there are still occasionally great anime, and there are certainly some anime I think are timeless classics (like Your Name, which might honestly be, in my opinion, one of the best films ever created), but I've got past the stage in my life where I always thought Japanese stuff was just better. A lot of creators from other countries now have taken influence from it and done really interesting stuff, and some of the best anime being made now is an international collaboration.