It's not that a country would not recognize the other citizenships;
Recognition of consular assistance rights is indeed one of the principal ways in which 'recognition' of the other citizenship would be meaningful, i.e. that's one of the things that I meant by saying 'recognizing' the other citizenship (recognize in a meaningful legal sense). And very much true, that if you're a citizen of eg Canada and you're visiting any other country in which you hold citizenship, Canada has technically no 'rights' by treaty for consular protection. (Sometimes other countries will allow as a courtesy).
(I believe consular protection/assistance rights under law is what would be the most current usage of 'diplomatic protection' referred to in that League of Nations treaty - although likely also cases where someone posted as a diplomat to a country while holding the host-country's citizenship would also not benefit from that diplomatic immunity, and possibly a few other corner cases not likely relevant here.)
But that's not the only context: some countries have, specifically, dual (multiple) citizenship statutes that outline specific rights, responsibilities, and restrictions about dual (multiple) citizenship - i.e. they do specifically 'recognize' dual citizenship under law, or sometimes only in certain circumstances, and this recognition (including permission to hold dual citizenship) might be limited to certain countries. Some countries might also only allow/recognize according to treaty (with certain other countries, usually friendly ones).
In this sense, Canada most certainly does not recognize dual citizenship, and I think that's true under US law as well. Canada doesn't forbid dual nationality, but for the most part it has no relevance to a citizen's rights/responsibilities vis a vis Canada.
In practice govt would have to 'recognize' that it can't provide consular assistance to a Canadian in a country where that citizen also holds nationality, and some other contexts, I suppose. Global Affairs does give consular warnings, sometimes, about issues that can arise for dual citizens, but that's basically just a courtesy.
Apart from the consular issue, the easiest way to boil it down is Canada doesn't care if you hold another citizenship, but the other country might - and it's up to the individual to figure that part out.
