Is the word Their boat a common or proper noun?

The Word 'boat' is a common noun; a general Word for any boat of any kind.
The Word 'their' is a pronoun, a possessive adjective, a Word used to describe the noun 'boat' as belonging to them.

The term 'their boat' is a noun phrase, a group of Words based on a noun that functions as a unit as a noun in a sentence.

Examples:

Their boat will pick us up at ten. (subject of the sentence)

They painted their boat a bright red. (direct object of the verb 'painted')

They bought a new sail for their boat. (object of the preposition 'for')

Note: A proper noun is the name of a person, place, thing, or a title; for example, "The Love Boat" (1980s TV series) or the Boat Street Kitchen (restaurant) in Seattle WA.