I'm not sure why but the lead goes crazy moving left and right when connected to animals. I found this while exploring and I tried to leash an horse. I tested to see if it was also with other leash-able mobs and it had the same effect.
Funny glitch although it should probably be fixed.
In the picture it does not show the glitch because it was going one frame to another so it was not able capture the effect well
MC-139374 Leashes glitching out