Basically, 'the' refers to a collective group. You see this often when we're talking about animals, and in other cases. For example: The cheetah is the fastest animal on land. We don't say 'a' cheetah because we want to talk about all cheetahs. I play the guitar. We don't say 'a' guitar because I can play any guitar, not just one. 'play the fool' is a similar, but slight different case. In ancient theater, 'the fool' was a character who mocked the royalty and academics by pretending to be stupid. So, to 'play the fool' means to pretend not to know something when, in fact, you do. It can also simply mean to try and make people laugh by acting silly. So, in that expression, 'the' refers to a specific, historical character, who was called 'the fool'.

To 'play the fool' is an idiom, meaning to behave in a silly, flippant way, or to waste time. 'The wolf' in this phrase is a general term, referring to no particular wolf, but wolves in general. For example: "The wolf is a creature that lives in the wild". Here the article 'the' is not referring to a particular wolf.

"Hungry like the wolf" is the name of a famous song by Duran Duran. So the use of "the" may have been influenced by the use in the song. However, when not singing along with the song, it would be more common to say "hungry like a wolf" or "hungry as a wolf." This is because you are not referring to any specific wolf, but to any wolf at all. It is similar to saying, "I am so hungry I could eat a horse." This is an actual idiom and it refers to any horse at all. "To play the fool" has already been answered. It was a phrase that became an idiom.