At many points in its evolution, Twitter's logo was very cartoonish [1]. Third party companies often seized on this to make the bird even more of a character on their websites.

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...

Twitter's character is the Fail Whale.