Good observation! I wonder if it’s got anything to do with the lolcats of yore (sidenote: I need to add “The Lolcats of Yore” to my band name list).
It’s sometimes hard to believe, given the proliferation of all the various types of memes nowadays, that there was a time that most memes were just funny pictures of cats in the form of a “demotivational poster”—I Can Haz Cheezburger? is a great example.
Perhaps the association of cats with the “old internet” is rooted in that, and when choosing mascots, sites like Neocities and Nekoweb wanted to choose one that reminded people of what the web could be like once more.