Yes, most animals can. Insects don’t breathe the way we do at all. They have separate openings for their respiration.

Amphibians start out with gills and only in adulthood do they develop lungs (althourh the axolotl keeps its gills for life) and can respirate through their skin.

Mollusks either respirate through their skin, gills, or through an air sack.

In none of those cases do the respiratory and digestive tracts cross. And those groups of animals outnumber us by far.