The ruling on the urine and faeces of a dog is the same as the ruling on all other impurities. What is required to purify it is to wash the item until the impurity disappears. As for washing it seven times, that applies specifically to the dog’s saliva only, as mentioned in the hadith.

Nothing in your house should be deemed impure unless you know for certain that this man touched it, and that his hands were still wet when he touched it and the dog’s saliva had not yet dried up, or he touched something wet when there was still a trace of saliva on his hands. For more details, please see the long answer.

Following the Prophet’s guidance, the impurity of a dog is to be removed by washing the vessel seven times, one of which should be with soil.