Yes and no. It depends on the animal.
DNA is composed of strand levels, for example, human DNA can be like: GGG-HHH-KKK-LLL-SSS.
Most animals, or all living things, will have GGG, and the closer an animal is related to human, the more strands it will have.
So, for instance:
Horse DNA: GGG-HHH-KKK-PPP-QQQ.
Human DNA: GGG-HHH-KKK-LLL-SSS.
You can combine the first three levels because they are the same. And in case you're wondering, monkeys/apes/primates have the exact genetic DNA code of humans.