Is pet a common noun or a proper noun?

Yes, pet is a common noun.

Nouns refer to people, places, and things. Nouns can be divided into proper nouns and common nouns. Proper nouns are names for specific people, places, events, and things, such as Rudolph, the Moon, October, and The Battle of Hastings, and are capitalized. Common nouns are nouns that refer to types of people, places, and things, such as microwave, reptile, trapeze, hilltop, decade, and dime, and are not capitalized except at the beginning of a sentence. Pet is a common noun because it refers to an animal. The name of the pet, such as Fluffy, is a proper noun.

Note: Pet can also be a verb, such as "Please do not pet seeing-eye dogs in training."