Is owl a common or proper noun?

No, the compound noun 'snowy owl' is a common noun, a general Word for a type of bird.

A proper noun is the name of a specific person, place, or thing; for example:

• Snowy Owl Drive, Silver Spring, MD
• Snowy Owl Inn, Waterville Valley, NH
• "Ookpik: The Travels of a Snowy Owl" by Bruce Hiscock