Can expresses a possibility, while does expresses something that either constantly is like that or something that is happening right now.
If you said: it can look like the politician, you would be expressing a possibility, so it isn't looking like the politician just now. If you say it does look like it, you state how it looks like now.
If in the second part of the sentence you were to use does, you'd be expressing that the puppet is sounding like the politician right now. Whilst if you say it CAN sound like that and that, you say it doesn't sound like it just now, but it could, theoretically.
Hope my answer is of use! :)