What generally accepted mnemonic phrases are there in Japanese that are used for memorizing a sequence of numbers or words?
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I'm not sure about "generally accepted"; some are probably more widely known than others. I would not be surprised to observe patterns corresponding to the 流派 (schools) of traditional Japanese arts, where the goroawase students learn can be linked to geographical location/study guide used/target school/etc.
I am personally fond of the variants on "naku yo uguisu" in your example (e.g. "haku yo uguisu", "the warbler vomits" for 894, etc.), and of course "Iroiro sankyū, Porutogaru" for 1639 when Japan's ports were officially closed to Portuguese ships.