Why does the scythe have a cooldown bar? Does it have a hidden ability?

When I use my weapons ability (in my case blocking with my sword) the red cooldown bar appears on my weapon and my scythe.

The scythe doesn't have an ability though, right? Is there something I'm missing about it? Why does it have a cooldown bar?

That's because all "swords" (as considered by the game) share the same cooldown (same with all hammers and/or daggers too). Yes, the scythe is just a sword with a strange sprite that is also flagged as a tool. They have a damage value, crit chance, everything just like a normal weapon.

Probably because it is flagged as a tool, the right-click ability and the Forge are disabled for the scythe, though. It has no special weapon ability but other tool-related ones (harvest some crops like kale and amaranth, and get hay from grass if you have a silo built).

There is no gameplay reason why this happens, and it makes no difference whether your scythe is on cooldown or not. Most likely, it is a programming artifact that is hard to fix, and it's probably not worth working on if it doesn't affect the gameplay in any way.