Are both Natsu and Igneel known as "The Salamander"?
Though IG_42's answer is essentially correct, it is complete.

From memory the conversation went something like this. Very simplistically. [Please edit if somebody has the subs handy :)]


  Lucy: I am a mage, looking to join a guild
  Natsu: I am looking for Salamander, Though he turned out to be human I am looking for a dragon 
  Lucy: (All surprised)
  Happy: Salamanders are fire lizard so we thought it might be Igneel


Natsu and Happy are very simple minded. It never occured to Natsu that someone else might be using his moniker to fake people. They heard rumours of Salamander and thought it might relate to Igneel.