There is nothing wrong with a hanging sign that swings. But you absolutely must not allow it to move sideways very much or it will beat itself up and even fall.
The problem is the chains—they are too long. The ideal is one link between eyebolts. This minimizes side motion. Or, another way is to use turnbuckles (that have closed eyes) between the two eye bolts. This allows leveling of the sign. The turnbuckle should be screwed in short or you will have the same problem as with chains. Also, S-hooks should always be completely closed, or use small clevis shackles. At the bracket you could even use swing set hardware bolted to the bracket, attaching the turn buckle directly to the swing set hardware, and then a clevis between the turn buckle and the sign's eye bolt. Use a clevis that can be locked with a cotter pin or a spring clip. A spring clip is like an industrial "bobby pin."
You can make the hanging sign rigid, but you must make sure your bracket is firmly anchored to the wall, otherwise wind movement can work loose the bracket where it attaches to the wall. Making the attachment rigid may also be a lot more complicated than it is worth.
I attach swinging signs all the time, but never with chains. Just a link, making sure the sure side movement is restricted.
Brad in Kansas City