No, the form he's is not possessive. The form he's is a contraction, a combination of the personal pronoun 'he' and the verb 'is'.
The contraction he's functions as the subject and verb (or auxiliary verb) of a sentence or a clause.
Examples:
He is the leader of the group. (He's the leader of the group.)
The hat that he is wearing is the latest style. (The hat that he's wearingis the latest style.)
The possessive forms of pronouns do not use an apostrophe:
possessive pronouns: mine, yours, his, hers, its, ours, theirs.
possessive adjectives: my, your, our, his, her, their, its.
The singular possessive form of "test" is "test's".
The singular possessive pronouns are mine, yours, his, hers, and its. The singular possessive adjectives are my, your, his, her, and its.
The singular possessive is ant's.The plural possessive is ants'.
Museum is singular. Museum's is singular possessive. Museums is plural. Museums' is plural possessive.
The singular possessive of dish is dish's
Taco's is the singular possessive.
The singular possessive form is classmate's.
The singular possessive is biker's; the plural possessive is bikers'.
The singular possessive is fish's.
Buzz's is the singular possessive.
Grave's is the singular possessive.
The singular possessive is Martin's.