The tree is rowan.
As i understand, the dog peeled off the bark today. What should i do to help the tree?
If i should use paint, is it possible to use bark-colored paint? Should it be a special paint?
Or should i use a special tree paste?
The tree will heal itself; anything you put on the damaged area will (in general) hurt, not help it. It will probably take 5+ years or so before the tree completely covers the wound, however. I once had a 7cm diameter tree get 85% of its bark peeled by a deer rubbing its antlers on it, and it survived.
I think your best course of action is in finding a way to keep the dog from further attacking the tree.