I’m not a biologist myself, so this explanation might be full of errors, but the gist of it is:

A dog got cancer. Cancer tissue is a part of your body with the same genetic information, just growing uncontrollably, right? Well, one of the cancer cells got separated and somehow learnt to survive on its own, reproducing asexually and started a lineage of a new single-cell organism that still exists to this day.

This single-cell organism has the same DNA as the dog. Genetically speaking it is a dog.