Yes. Ichigo's hard work, determination, and the accumulation of power and experience allow him to become a captain in the manga. It's a significant milestone in his story arc.
Falcon doesn't become Captain America in the comic universe. The mantle of Captain America has its own established history and succession that doesn't include Falcon.
In the comics, Falcon doesn't take on the role of Captain America. The Captain America identity has had various successors, but Falcon isn't one of them. This is part of the complex and evolving storylines in the comic universe.