Putin wants to destabilize. He's not trying to shape politics one way or another. He wants chaos and anarchy for his enemies. However, lots of billionnaires wish to influence society toward their "vision".
He’s a conservative foreign billionaire and influences US politics (which addressed the parent’s question in my opinion).
And he’s certainly trying to “shape politics one way or the other”. The whole disaster is in Ukraine is based on shaping geopolitics. And US conservative politicians and pundits have been used in the run up for that purpose
He cares about the old Soviet Union/Russian Empire glory days. That's it. He could not care less who's in the white house, he'll cause them trouble whoever they are --that's all he cares. He personally may be conservative but he is not trying to make the US conservative or liberal --it makes no diff to him.