TL;DR - because rates are a specific form of tax under common law, and the nomenclature comes from the history of the rates in the colonial Hong Kong legal system, which was related to police pay.
Disclaimer: I am not a qualified financial adviser nor an expert in financial law.
The first Rating Ordinance, Ordinance No.2 of 1845, was collected with the express purpose of paying the expenses and upkeep of the police force.
The Hong Kong police at that time was called the 香港差役. with 差役 referring to a bailiff sent from a 衙門 yámén. Hence the 差 is pronounced chāi in Standard Mandarin and caai1 in Cantonese, and refers to being "sent out" or "dispatched".
Thus, the provisions for these forces were given the name 差餉 caai1 hoeng2, and equated to the British version of the common law concept of rates.
Note that Singapore and Malaysia follow US custom in calling it property tax and 房地产税, whereas Macau, Taiwan and PR China all use civil law systems and do not have the common law concept of rates.
Also, it must be said that like in the UK, property tax and 物業稅 are not the same as rates and 差餉 in HK; the former includes the latter.
…… 差饷是香港对地税的说法，承袭自中国的古文。中国内地并没有差饷。差饷是一种源于英国的税款, 在英国、美国、澳大利亚以及很多国家和地区都存在 ……
It's not from ancient Chinese. It's a translation of English word (maybe rate), though I don't know how it was translated, seems only circulated among Hong Kong region. 差餉 to rate is just kind of translation back.