Isn't that more of a logo than a character? I'm not really sure where you draw the line, but it doesn't feel like the bird has been much more than a logo for Twitter.
People would also often reference the bird, because of the connection that birds "tweet".
I can't say if they used the bird extensively in their documentation and backend tools, but that's where it's more likely to have been playfully included since their frontend was always pretty simple and to the point.
[1] https://www.imore.com/sites/imore.com/files/styles/larger/pu...