I presume it would only be a bad move if you hold some inflammatory views that could be traced back to you or some views that would leave you in an uncomfortable position socially/personally.
A wise person really does not speak to the media. How many people do we all see on the news being chased by a cameraman and just constantly replying "no comment"? As I'm sure you're aware, things can be twisted to suit any story; items omitted to make views appear biased etc.
I'm unsure on the issues of recording the conversation and confidentiality, but I feel that the proper answer here lies extra-judicially. Newspaper reporters thrive on their sources. There are many examples of journalists being imprisoned for contempt of court for failing to disclose names from which they got information and such. If a journo is seen to be compromised, nobody will talk to them, nobody will trust them and that means the journo has no realistic future career. You should make it clear when you speak again that if you are compromised, you will see to it that the journo's reputation will be severely tarnished. I'm not advocating threats though! The pen is mightier than the sword.
If you're not a public figure then I doubt the journo has anything to gain by releasing your name. Usually this is only done for shock value and in cases of political significance.