Why does 千秋 mean a swing?
in 唐 dynasty (618-907), there're records of 千秋:

the peom 樓前 by 王建

天寶年前勤政樓
每年三日作千秋
飛龍老馬曾教舞
聞著音聲總舉頭


another one is the 漢武帝後庭鞦韆賦 by 高無際, which stated:

況鞦韆者﹒千秋也﹒
漢武祈千秋之壽﹒
故後宮多鞦韆之樂


later, in 記纂淵海　卷八十九, by 潘自牧 (~1195) of 宋 dynasty (p33-34 of pdf file);

https://archive.org/details/06065623.cn

that the term "鞦韆" was elaborated. mr 潘 claimed that, according to mr 高:

鞦韆﹒漢武帝後庭之戲﹒
本云千秋﹒祝壽之詞也﹒
語訛轉為秋千﹒後人乃旁加革





briefly, it's 千秋  --> 秋千 --> 鞦韆

back to the original question, the game "swing" in cantonese is "千秋", which share the meaning "thousand autumn", a term for wishing for longevity.

lastly, i would say that it's one of the numerous evidences, that cantonese is highly correlated with middle chinese. or, one might say that cantonese preserve more / change less than other "dialects".

have fun :)