A doppelganger that's changed shape and detecting thoughts radiates two magic auras
Supernatural Abilities says
Supernatural abilities are magical but not spell-like. Supernatural abilities are not subject to spell resistance and do not function in areas where magic is suppressed or negated (such as an antimagic field). A supernatural ability’s effect cannot be dispelled and is not subject to counterspells.
(Also see Player's Handbook 180, Dungeon Master's Guide 289, and Monster Manual 315.)
Thus on the first round a patron uses an effect like detect magic and catches the disguised mind-reading doppelganger in the effect's cone, the patron will detect the presence of magic. Then, on the second round, if no other auras are present, the patron will detect two auras, with the DM determining which aura is stronger. (While the doppelganger's supernatural ability detect thoughts is the equivalent of 2nd-level spell, the doppelganger's supernatural ability change shape isn't given a spell level, although the alter self spell seems a reasonable comparison to this DM.) On the third round, the patron knows the auras' precise locations and can try to identify the auras' schools.
It will be up to the doppelganger to either notice the concentrating patron and subtle try to disrupt his concentration or explain away the magic auras of divination and illusion as ongoing tools for keeping his inn safe from intruders and looking nice, respectively.