Should I use or replace an existing ceiling fan bracket?
I agree that what we are looking at is an old bracket for mounting a box for a light fixture. Here is the Westinghouse version of a retrofit bracket which you can find in most hardware stores.

I can also tell you from experience, that it isn't very easy to install. Wait until you try and mount the box to the bracket and you will see what I mean. If you can get to it from above its much better. If you want to cut a larger opening and install a different type of bracket you can use a medallion. To cover the hole instead of a patch and paint.

They come in different sizes and decor. That's all I've got, good luck.