Yes, i clicked wayback link and seen that it was moving when i hovering over. I was thinking it is moving in specific time frame. and guessed it's a GIF.
No, it is not an animated gif; it is an HTML5 animated banner. Animated gifs lack a mouse-over function; they simply loop the animation continuously.
do you know reason for banning the animated signatures? In gifs we can say it takes more storage and when uploaded by many users it will take huge amount of storage.
and it's not annoying too, because we will see flashing lights when we hover on it.
As far as I know, there have never been animated signatures. Signatures are made using BBCode, and since BBCode does not allow any form of animation and GIFs in signatures are not enabled by forum admins, animated signatures are not possible. What you see on the Wayback archive is not a signature but an HTML banner (you will notice that it is displayed below the post area) that were active on the forum at that time.
Regarding animated avatars, I am not sure why they were removed, but it probably wasn't due to storage issue.