"You say you know me inside and out, but I'm not like a radar in that sense. Who I am is the key, what I am is the answer."
To know something inside and out is to claim thorough knowledge of it. Things people often claim such knowledge of: Themselves, their job. a hobby, a sport, etc.
Radar - A system of transmitted and received signals used to indicate presence, distance, direction, etc of aircraft and ships, weather and so on. Also, a character from MASH, known for detecting incoming helicopters before anyone else. Neither one known for being able to see inside something and therefore know it 'inside and out'.
Who I am is the key - Clever answer to the riddle says "The Key". Doesn't satisfy the other parts of the riddle. Keys unlock things but keys are rarely referred to as 'who', more often a what. A key can also be a chart of correct responses, i.e. an Answer Key; The pitch of a musical piece, an adjustment one makes in an activity, a guide or explanation of symbols as on a map, parts of a typewriter or keyboard, and 40+ other things.
What I am is the answer - Could be a dead line just to end the riddle since we already know we are looking for an answer. Indicates the answer is probably a thing.
Knowing Inside and Out does not correlate well with it not being radar since radar does not do this anyway. There is not the chance, however remote, that a listener would have, momentarily, thought something related to radar.
You say your DM gave you the riddle. Possibility presents itself that the DM is the subject of the riddle. Replacing the pronouns with DM does not immediately break the riddle.
Also possible to replace the pronouns with the word Map. A map can be known inside and out, fulfills a similar purpose, though not identical, to radar. A map generally contains it's own key to reading the symbols on it while also being a key of sorts itself. Strong possibility.
Other considerations:
As a gamer, it is possible to know an adventure or module inside and out. Adventure does not fit well with radar, but might be made to at a stretch. The last two lines would have to be near throw aways though, I feel.
What is your GM's name? Strong possibility this may have some bearing on the eventual answer. Especially if he fancies himself to be too clever by half.
My best guess at this stage, with nothing else to go on, is Map.
