"give him a bath" and "bath him" mean near enough the same. "bathe him" has a similar meaning, but to me sounds more specialised and less like a routine bath. "shower" and "bath" are two different ways of cleaning one's body.
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Mister Micawber'Bathe' is relatively rare in AmE; in BrE it means 'swim', I believe.Right, that would be an intransitive meaning though.