From an animation standpoint, it's not on the same level of influence as Snow White and Toy Story, but from a comedy level, it's much more influential.
With the exception of The Simpsons, no animated film or series, that in theory should have been aimed at a youth audience, ended up being more loved by adults. This is mainly due to it's clever references in popular culture.
Snow White and Toy Story are probably better films in terms of works of art, but if I had to watch one of them as piece of entertainment, I'd watch Shrek because I would spend 90 minutes laughing my arse off.
I think it did, in several ways. First off it made DreamWorks a serious contender in the animation field, and gave Pixar it's first competitor. Shrek made DreamWorks Animation big.
Second of all it proved that adults could enjoy animated movies just as much as children, if not more. Up til then animation was mostly seen as something for children or families. Shrek put all of that to the test.
It also brought something new to the animated fairytale genre. A fresh breeze.
In many ways, yes. While Shrek was far from the first Animated Movie to be in CGI (which was Toy Story) or to include Pop Culture references (which was Aladdin), Shrek was influential enough to change the course of Animated Movies. In many ways, it made people and moviegoers eager for movies that were zingy, zappy and zany and had Pop Culture References. As Hand Drawn Animation almost completely disappeared and most animated movies that followed afterwards were completely influenced by Shrek. As they were comedies that were Pop Culture driven and had Celebrity Voices (which was a trademark from Shrek). True that Shrek wasn`t as invigorating as both Aladdin and Toy Story. But it did made it`s impact on it`s own right, which was clearly seen by the animated movies that followed afterwards. So to deny that it had an impact is (I`m sorry if this remark comes across as condescending, as it wasn`t my intention) pretty naive.