1)By using the duck analogy; the speaker is basically saying 'It looks like a prison, and functions like a prison ,despite them calling it a clinic, so therefore it is a prison'. The title given to it does not change the intrinsic quality of it being a prison' 2) Deductive reasoning. If I hear a baby crying, I think a baby is making that sound. However it can be used to qualify a weak argument, where you do not have all the facts to hand and want to justify your conclusions to another. You are jumping from one piece of evidence (hoofbeats) and leaping to a conclusion (zebras) without all the facts available.(You have not seen that it is in fact zebras). In this case it is also being used as an analogy I believe.

Question 1. The writer feels that the place is like a prison, or is a prison despite the fact "they" say it is a clinic. Question 2.It appears to be a duck, so you conclude it is a duck whatever anyone else calls it. When you hear hoofbeats... is new to me, but a google search suggests it means think the obvious (horse) rather than the unusual (zebra). The two expressions are not the same.

'If it looks like a duck and swims like a duck, then it is a duck' simply means that you can try and obscure the true identity of something but it's true identity will become known by observation of the thing. It's also common to replace 'swims like" with 'quacks like' In the field of Logic this phrase can be used in Abductive Reasoning.