As @Brian THOMAS alludes to in a note, you do recognize intervals - in the case he mentions, the octave leap in Happy Birthday. Or consider the famous third in the opening of Beethoven's Fifth Symphony. Or in fact the fifth in the opening notes of Twinkle.
Perhaps the issue here is not so much recognition as nomenclature - we use user-friendly names for tunes (like HB and Twinkle) but notions of fifth, octave, minor second, etc. which take take a bit of explaining (i.e. basic interval theory).