"That doesn't sound like him."
ONE PARSE: That/doesn't sound/like him
A deictic pronoun.
It refers to: Discourse Deixis
Discourse deixis is deictic reference to a portion of a discourse relative to the speaker's current “location” in the discourse.
The speaker is in effect "pointing with his/her utterance to something outside of it. It refers to something heard.
verbs of perception in English: look, sound, feel, taste, etc.
to sound=verb of perception
In English grammar, a verb of perception is a verb that conveys the experience of one of the physical senses. A few examples would be see, watch, look, hear, listen, feel, and taste. A verb of perception is also called a perception verb or perceptual verb. Distinctions can be drawn between subject-oriented and object-oriented verbs of perception.
verbs of perception
doesn't sound like him = is the sentence's predicate.
If the sentence stated: "He sounded angry." "sounded angry" would be a subject complement.
Predicate means "The part of a sentence or clause containing a verb and stating something about the subject (e.g. went home in John went home)."
like is used to compare two things:
This cake tastes like your cake.
Like as a preposition meaning ‘similar to’ [bolding mine]
Like means ‘similar to’. We often use it with verbs of the senses such as look, sound, feel, taste, seem**: [bolding mine]
use of like in English: Cambridge Dictionary
like him is used to compare the deictic subject that with the person referred to by the pronoun him. "like him" is a prepositional phrase.
When like is used as a comparison, it is the first term of a prepositional phrase: like him, like them, like your brothers, for example. Like is not an adjective here as the dictionary shows. It is used to show comparison.
the verb is: to sound, the negative third person singular is /doesn't sound/.
Clause: In language, a clause is a part of the sentence that constitutes or comprises a predicate. A typical clause consists of a subject and a predicate, the latter typically a verb phrase, a verb with any objects and other modifiers.Clause defined in Wikipedia
SECOND PARSE: That/doesn't sound like him.
That is a single clause sentence consisting of a subject, predicate and prepositional phrase that describes what he sounds (doesn't sound like here) like and is therefore adverbial.