Cancer tissue is a part of your body with the same genetic information, just growing uncontrollably, right?

I have no idea. Don’t ask me to verify this stuff!

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.

I regret asking!