Dog Style is the same as "Doggystyle", however, it is a more appropriate term because "Doggy" is somewhat of a misnomer. Children call dogs "doggy", but adults (who have the actual sex) would refer to dogs as "dogs."
Woman: Phew! That was a real reaming!
Man: Yeah! Should we try to go right from reverse cow girl into dog style?
Woman: Boo yeah! I'll get on all fours!
Man: I'll enter from behind. I love dog style.