A scale plays, you name it. Recognizing scales and modes by ear helps you improvise over the right notes, understand the mood of a song and hear what makes a melody sound bright, dark or exotic.
Level 1 is major or natural minor. Harmonic minor joins at level 2, Dorian and Mixolydian at level 3, melodic minor and Lydian at level 4, and Phrygian and Locrian at level 5. Levels 6 to 8 then add the major pentatonic, the minor pentatonic and the blues scale.
Compare each mode to the major or minor scale it's closest to and listen for the one note that changes. Dorian is minor with a brighter sixth, Mixolydian is major with a lower seventh, Lydian is major with a raised fourth. That single note is the fingerprint.
No. Knowing the names helps you talk about what you hear, but recognizing the sound comes first. Many players learn to hear a Dorian color long before they can spell the scale.