As you well know, this is a common Cantonese colloquial phrase which Mandarin speakers will have a hard time appreciating.
We use it all the time to express a situation where there is confusion / disorder / a melee of riotous activities. An English equivalent would be something like "pandemonium" or "chaotic", not necessarily in the context of a sporting contest, but includes a shouting match of people quarreling.
So, the extended meaning of 打 here would include "against", "versus", "oppose"
Confining it to just the physical meaning of "hit" or "beat", do not do justice to this really colorful and useful phrase. I always keep it handy when I have no good answer to any embarrassing situation or wanting to politely get rid of a pesky busybody.
I am not conversant with any historical, literary usage, but in the context of everyday Cantonese usage, interpreting it as "is" is odd, because the whole phrase is not about "who is who", but who is doing what to whom, hence 打, (hit or beat)
My knowledge of Cantonese at the best of times is limited to the "rough", colloquial variety, so please take my answer in that spirit.