Ich liebe seinen Sohn.

English Translation

I love his son.

Why is it ‘seinen’ (accusative) rather than seiner (genitive)?
Why is ‘his son’ not possessive?

Because it is “jemanden lieben”.

Of course, “sein; here: seinen” is a possesive pronoun but the case depends on the verb.

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I looked up possessive pronouns. I think (though I admit to being no expert) that you may have meant that seinen in this sentence is a possessive adjective.

‘Seinen’ would be correct for a masculine possessive adjective in the accusative case.

You can say either possesive adjective or possesive pronoun (in adjective use) or even possesive determiner.
Possessive - Wikipedia (chapter Terminology)

Anyway, you got exactly what I tried to explain.

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