In the Dragonlance setting, can a dragon recognize that a player character is a wizard, if they are a 2nd-level wizard trying to hide that fact?
Wizards are not different to anyone else, as they don't have an inner power source, but manage the magic via Solinari, Lunitari or Nuitari, just like a cleric would. As long as said wizard isn't keeping any spells up, like Mage Armor, or doesn't wear anything magic that can identify him as a wizard, like a magic arcane focus or a spell-protected tome that the dragon can "Detect Magic" on, your wizard is safe.
The dragon can, though, deduce it. Remember that they are supposed to be keenly intelligent and have better senses than humans. A Dragon could smell the special ink wizards use to write on their tomes, or even see traces of it in their fingers; maybe smell traces of some common material components; he could maybe deduce it from clues in a conversation; it could cast a mind reading spell or ability; or it could have spies or previous information on the wizard.
That being said, no, Dragons do not have any specific ability that lets them detect if a mortal is a wizard or if said wizard has passed the Test or not. They can still deduce or investigate it, like any other intelligent creature would.