Ah! A Dutch cap originally was a cap, Mr P. I assume it became attached to the contraceptive device because of the similarity of appearance as well as the 'mickey-taking' aspect. Other such devices also have the same critical element between rival nations...we refer to 'French letters' and the French refer to 'English overcoats', for example. Neither nationality is talking about mail or garments!
The earliest recorded use of the cap, in the sense you use, it was in an article by Marie Stopes in The Lancet in 1922. Cheers