First, in every single lyrics website that I've checked, it is the hunt, not a hunt. (Overall, these websites contradict each other as to the lyrics: some websites omit the word down, and one finds other discrepancies (the earth is a fire vs the earth is afire, etc.)
Next, the band/songwriter(s) wanted to use the definite noun phrase the wolf in the simile hungry like the wolf instead of an indefinite noun phrase (a wolf). Since we cannot read the minds of the songwriters, we can't say or explain why they made this choice. The alternative hungry like a wolf is grammatical and, in at least one sense, the expected phrasing.
We don't even know if the wolf refers to a 'generic wolf', an actual wolf, or some other kind of wolf. Apparently the lyrics are based in the story of Little Red Riding Hood.
However, we do have the text as it is, and the text is Hungry Like the Wolf. This is used not only in the chorus, but as the title. In short, to state the obvious, hungry like the wolf uses a definite noun phrase, which (by definition and by description) is definite as opposed to indefinite.
Sometimes, the use of definite language is greatly desired over the use of indefinite language. For instance, if a gal I'm interested in says, 'Let's go out to a movie sometime', that's indefinite and I'm not going to set my hopes on such a statement. But if she says, 'Let's go see the 9:30 showing of Title of Movie on Wednesday of next week', she's speaking definitely and that's a date, as we say, set in stone.
In the song, using hungry like the wolf is simply definite versus the indefinite hungry like a wolf. One could use a wolf throughout the song. The same for the forest in the line stalked in the forest too close to hide. One could use a forest there as well. Just as 'the forest' talks about a definite forest (such as the one the hunted person is in), 'the wolf' talks about a definite wolf, such as the one that is hunting the person. In this sense, this song is about a definite forest and a definite wolf. Since, grammatically, that which is definite is also specific (but note in passing, not vice versa), the song is about a specific forest and a specific wolf. In narrating a story, if we read this song as a narrative, the use of specific items makes the story 'up close and personal'. It's not just any old forest we're talking about, but the forest, and it's not just any old wolf, but the wolf that's pursuing you. (The song gets even more up close and personal with its choice of other determiners: 'your skin' and 'my heat', for example.)
Or consider another example of the difference between a definite and indefinite noun phrase: you and I are standing on the edge of a forest and I suddenly yell: Watch out for a wolf!, well, you might take me as being mad, since in that context, that statement has little applicable meaning. You might ask me what I mean and what wolf I'm talking about. But if I suddenly scream Watch out for the wolf! you're probably more apt, if you're a native speaker, to perceive you are in immediate danger of encountering a definite wolf. You might ask 'What wolf?' but by then it might be too late.
I said earlier we might expect the wording hungry like a wolf. This is because it parallels such popular simile-idioms as hungry as a bear and hungry as a church mouse. In fact, a Dictionary of Similes (1916) lists a dozen other such similes, including hungry as a wolf.
SIDEBAR:
Pointing out that the wolf can be a generic noun phrase doesn't say much, since (a) we don't know whether Duran Duran is using it as a generic noun phrase, (b) there are three types of generic noun phrases, and both a wolf and wolves can be generic noun phrases (see Re: A question about the generic use of articles, by John Lawler, and (c) it occurs in a simile, in symbolic language, in a song, inspired by the tale Little Red Riding Hood.