My two cents.
1 I don't share his opinion.
2 I don't have the same opinion as he has.
3 I don't support his opinion.
1 and 2 mean exactly the same thing, or as near to the same thing as two different sentences can mean.
3 means something different, and I think what it means is
somewhat nonsensical.
An opinion is a belief or judgment. To support, in the definition
which most closely fits the context, is to aid the cause of by
approving, favoring, or advocating. One may support a war or one
may support a charity devoted to curing cancer, and so on, because
these are causes. They are undertakings. A belief is not an
undertaking. A judgment is not an undertaking. I don't think,
therefore, very strictly speaking, that "support an opinion" is a
well-formed expression.
Relaxing our standards a bit, we'd have to say that when we "support an
opinion", we approve, favor, or advocate it. We are willing to
argue in favor of that same belief or judgment.
CJ