Here, the new cube is selected but not active. Probably Alt+D additionally activates it whereas your code doesn't .

But it works the first time.
Also, I couldn't find an 'activate' method. Is that internal?

If you delete the first copy, there is no active object. Hence, after creating the second copy, the copy is also not activated.

It's the same if you do it manually: In a new scene, delete the lamp and the camera. Select all. Alt+D. Delete the new cube. Select All ("A" key, do not select the cube). Press Alt+D again. Now the new copy is also not selected.

This doesn't seem like correct behavior, since the first time it works. I changed the method to be duplicate (Shift-D) and get similar results. I also added the print(current object) -- So you can see it selects/activates the new object the first time, but not the second.
However, if you use the mouse to select the original cube, then when you run the script the SECOND (or third) cube is selected/activated.

Should I do something in addition to bpy.data.objects["Cube"].select = True
in order to make it work as expected?

bpy.ops.object.duplicate_move(OBJECT_OT_duplicate={"linked":False, "mode":'TRANSLATION'}, TRANSFORM_OT_translate={"value":(-1.43941, -0.658703, 2.10968), "constraint_axis":(False, False, False), "constraint_orientation":'GLOBAL', "mirror":False, "proportional":'DISABLED', "proportional_edit_falloff":'SMOOTH', "proportional_size":1, "snap":False, "snap_target":'CLOSEST', "snap_point":(0, 0, 0), "snap_align":False, "snap_normal":(0, 0, 0), "texture_space":False, "remove_on_cancel":False, "release_confirm":False})
print(bpy.context.active_object)

It's by design that the copy is only made the active object if the original was the active object. This is true with the first time making a copy. It is not true with the second time because no object is active when making the copy.

What you can do is, after deleting the first copy, activate the cube again. Then making another copy activates the copy.