Does Xbox 360 Recognize Achievements for Halo 2?

Yes, but only for Halo 2: Anniversary (the remastered version in Halo: The Master Chief Collection).

The original Halo 2 (2004) on Xbox 360 has no achievements.

Why?

• Halo 2: Anniversary released in Halo: The Master Chief Collection (Xbox One / Series X|S) with a full achievement list.
• The original Halo 2 predates the Xbox 360's Achievement system (introduced with the 360 in 2005).

Playing the original Halo 2 via backward compatibility on Xbox 360 yields zero achievements. To earn them, play Halo 2: Anniversary on Xbox One or Series X|S.