Is this a verb phrase or simply an adjective?

"The quote is usually attributed to him."

The citation above that you use contains a verb phrase in the passive voice, not a predicate adjective complement, meaning it is not adjectivally saying "the usually-attributed-to-him quote" but is conveying an action that unnamed subjects usually perform, namely the action of attributing the quote to him. For the sake of illustration, if we were to add "by people" to the end so that the performer of the action is no longer unnamed, then the active voice version of the sentence would be "People usually attribute this quote to him."

Citation from KissGrammar.org:

Having studied the passive voice, you may have wondered if some of the sentences can be considered simply as S/V/PA [Subject/Verb/Predicate Adjective] patterns. Sometimes, they can be. Consider:

1. He was worried about the game.
2. The Eagles were defeated by the Patriots.

In (1), "worried" describes the emotional state of "He" more than it denotes any particular action. But in (2), "were defeated" denotes a specific action performed by the Patriots. Thus some grammarians would consider "worried" a predicate adjective, whereas "were defeated" should be considered as passive voice. In effect, the two constructions, S/V/PA and passive voice, slide into each other, and thus how you should explain it may depend on how you interpret the sentence.

The quote is usually attributed to him.

No: there's no subject complement. The verb phrase is the passive "is usually attributed to him", consisting of the verb "is" followed by the past-participial clause "attributed to him", functioning as complement of "is".

Note that this is called a 'short' passive due to the absence of a by phrase.