Well, if you die of something else, these terms are not used. Or maybe I didn't see, but you don't say 'after fighting alzheimers' 'after fighting ALS' etc, they just say; he/she died of X. That feels more fair than taking some kind of process into account whereby one gets the feeling that if you had battled harder you would've come out on top. For many cancers this is simply not the case yet; I was 'lucky', he was not, but that's all it is. It has not much to do with fighting IMHO.
I wonder if one reason for the difference in language is that chemotherapy is so brutal that it ends up feeling like a "battle". Plus maybe that there's a chance of "winning", which isn't there with Alzheimer's or with ALS.
(I'm not justifying the correctness of the word, just speculating on the cause.)
It's that "chance of winning" thing when "battling" that's part of the reason it shouldn't be used. Too easy to imply (or for others to infer) that if they don't win it's because they didn't fight hard enough. Better to just get rid of the battle analogy altogether, IMO.
I've read phrases like "battling", "fighting", etc. used in a variety of illnesses. Its never been apparent to me that it is specifically relating to cancer.
Might come across as snarky, but really meaning not to: What's an expression that would be more acceptable?