There is no difference in meaning. 'Sort' is somewhat less formal than 'kind', and 'type' is somewhat more formal than 'kind'.
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Mister MicawberMany types/sorts/kinds of handbag(s)/animal(s)/house(s). After "many types/sorts/kinds of" ,the following noun should be plural or singular -- It is better style to use singular + singular ('a kind of dog') and plural + plural ('3 kinds of dogs'), but the others are not absolutely wrong.From your context,what is meant by the others are not a