I have an animation in blender, I want this animation copied onto an other object, that it does the exact same thing as the other object. (the same animation)
Assuming it's just a key framed object and not a rigged object, your question was answered sufficiently here I think but I'll copy over Richard Marklew's answer just in case the link dies some time in the future but the hotkeys are still the same by then: "Select the cube with no animation, shift select the cube with the animation then Ctrl+L / Animation data (or Object / Make Links menu) to link the animation. Alternative you can use the Dopesheet set to Action Editor to apply any action to your object any action. You can then press the number to the right of the action to make it a single user so the animations are then no longer linked (or U in object mode)"
I will add also that you can accomplish a dynamically updating version of this with constraint modifiers.