Kang joined Pohang Steelers in 2014 as a free agent rookie and go on to make his debut in a AFC Champions League group game on 18 March 2014 against Shandong Luneng Taishan in a 2–2 draw, where he came on as a late substitute.[1] This would be followed by his first starting appearance, which was against Jeju United on 5 July 2014, in a league game that ended in a 0–0 draw.[2] He would become a regular within the team until he was loaned out to Sangju Sangmu, the sports division of the Republic of Korea Armed Forces, as part of his military service.
On 15 April 2022, Kang joined Chinese Super League club Beijing Guoan.[3] He would make his debut on 5 June 2022, in a league game against Cangzhou Mighty Lions that ended in a 2–1 defeat.[4] After his spot on the team was given to Samuel Adegbenro ahead of the 2024 season, his contract was terminated by Guoan on 6 March 2024. He scored 13 goals and provided 12 assists across 59 games in all competitions during his time with the team.[5]
On 7 March 2024, K League 1 club FC Seoul announced the signing of Kang. He was given the number 15 shirt.[6]
^Zhang, Kunlong (6 March 2024). "聚焦|国安官宣姜祥佑解约离队 重返K联赛未来两天将有定论" [Focus: Guoan Announces Kang Sang-woo Departure, May Return to K-League Soon]. www.163.com. Retrieved 6 March 2024.