Greece international Giakoumakis, 28, tallied 17 goals and three assists in 27 matches (21 starts) this season.
The award recognizes the year's most impactful player with prior professional experience who made his MLS debut in 2023.
Giakoumakis signed with Atlanta from Scotland's Celtic in February to replace Josef Martínez.
Messi, who joined Miami in July, was nominated for the award despite his appearances in MLS league play being limited by a leg injury suffered while on duty with Argentina in September.
He was able to play just 372 minutes in six appearances (four starts), scoring one goal and providing two assists as Miami missed out on a place in the MLS playoffs.
Messi, who won his record-extending eighth Ballon d'Or award on Monday, still had much to celebrate from his first months in Miami.
The Argentine World Cup winner scored 10 goals in seven games to lead the club to its first trophy with the lifting of the Leagues Cup. He also played in Miami's U.S. Open Cup semifinal victory over Cincinnati.
Reuters contributed to this report.