Having dual citizenship simply means, well, you have two citizenships. If you are both a US and a Canadian citizen, you are simply entitled the rights of both American and Canadian citizens, and held to the duties of both. Nothing prevents you from having more citizenships - if one born to Japanese and Canadian parents in the US, one has Japanese, Canadian and American citizenships (technically only before 22 though - Japan does not recognize multiple citizenship for adults and requires a selection before one attains 22 years).
However, citizenship is essentially regulated by domestic law, where international treaties are only a secondary regulatory factor. Even if the hypothetical person described above declare that he/she gives up American and Canadian citizenships to Japanese authorities, American and Canadian authorities could not have and will not acknowledge this declaration. As far as American and Canadian authorities are concerned, that person is still a citizen. He/she may still renew his/her US passport and vote for president if he/she wishes.
In this case, it is just that Japanese authorities consider him/her a Japanese citizen only and will treat him/her as a citizen. For example, Japanese authorities will not be obliged by the Article 36 of the Vienna Convention on Consular Relations to report to US consular officers had the person been arrested in Japan - because that person is not a US citizen as far as Japanese authorities are concerned. Otherwise, if he/she manages to never get arrested and present only to Japanese border control his/her Japanese passport, there is really no reason he/she will get into any trouble. A state's non-recognization of multiple citizenship only concerns itself, and have no impact on the government of foreign states.
PS: I have heard of stories where Chinese citizens naturalized in Portugal have their Chinese passport confiscated by Portuguese authorities (due to their naturalization) and then report to Chinese authorities that their passports are "lost" so that they get new Chinese passports. Portugal accepts multiple nationality, but in recognition of China's non-acceptance, they do confiscate Chinese passports should a Chinese citizen naturalize. Portuguese authorities acknowledge that the person have forfeited his/her Chinese citizenship, but Chinese authorities do not (because they did not register the forfeiture of their Chinese citizenship with Chinese authorities) so the Chinese embassy still considers him/her a Chinese citizen.
Note that this is not legal - failing to report naturalization to Chinese authorities is misrepresenting facts to authorities and will result to the least the stripping of one's Chinese nationality. However, this will happen only if they actually find out - the Portuguese government has no obligation to report naturalizations to Chinese authorities.