Nope. Conan will not give away kid's identity. I'm sure everyone already understand him very well to know that if Conan want to catch kid, he will catch him in the act. If kid was actually around but he is not there for a crime, then Conan will turn a blind eye to him. Moreover, what evidence can Conan give to say that his true identity is kid? He can't prove kid is him!
However, if he found out that kid had a reason to steal, he might actually help kid take a lookout for him. And if Conan found a clue, he might actually send a notice to kid. The idea of them helping each idea May actually prove to be interesting ... Kekeke