Norway striker Haaland scored 52 goals in 53 games to help City win the treble of Champions League, Premier League and FA Cup, while Bonmati inspired Spain to World Cup glory earlier this month.
Haaland's achievements in his first season at City after joining from Borussia Dortmund saw him take the men's award, for which his club colleague Kevin De Bruyne and Lionel Messi were also shortlisted.
Messi was at Paris Saint-Germain last season but has since left to join Inter Miami in Major League Soccer.
"I am kind of living a dream. This was my dream when I was young, so to be able to do this together with my teammates is something special," Haaland, 23, said after collecting his award.
"It was a season I will never forget," Bonmati said.
Spain coach Jorge Vilda missed out to England manager Sarina Wiegman for the women's coach of the year prize.
Pep Guardiola won the men's coach of the year award after leading City to the treble.