Nothing wrong with the sentence, actually. First, it is syntactically well-formed. Here is a tree drawing:
It is just one where there are two logically possible bindings for the possessive pronoun his. k=i or k=j. So if you want to say something about the sentence, it's semantically ambiguous. A similar case would be sentences like
Thomas reminded Billy to put his toys away.
Billy asked Thomas for help with his homework.
The two examples aren't a problem because you understand pretty quickly what his binds to. The OP's example, you also understand quickly that people two generations apart don't fight in the same battles. But it's funny to think about someone fighting alongside their grandfather in 100 years ago. Very funny.
The grammar school books will have you believe that there is a problem with ambiguous sentences that have an improbable but amusing reading. Maybe they picked this example because it seemed it would be parsed on the fly as involving a parallelism of possessors, leading to the wrong semantic reading.
The bigger problem for students of English is writing ambiguous sentences that have no clear resolution, rather than an improbable but funny one. Avoiding it takes practice writing and reading rather than learning slogans, because it's silly to avoid all ambiguity when writing (this is not computer programming). Then again, if you diagram out every sentence you write and record all of the bindings (pronoun to antecedent), just to make sure that there is no ambiguity, that is probably good practice!