The new decree of November 14, 2022 contains an amendment (bottom of p2) to the Federal Law 53-FZ originally enacted on March 28, 1998, concerning military service. The former version of the text contained separate provisions in Article 2 about different kinds of military service that could be performed depending on nationality. This allowed Russian citizens who also held another nationality, or foreign citizens, to serve in the Russian armed forces under contract, and only in non-officer ranks. Russian citizens without other allegiances were able to serve voluntarily, or be conscripted, and were additionally eligible to serve in the state intelligence apparatus and other bodies, as well as in the main armed forces.
The amendment is slightly differently organised, but the key part is the text which extends service "по призыву" (on demand) to "гражданами в том чсиле имеющими гражданства (подданства) иностранного государства" (citizens, including those who have citizenship (nationality) of a foreign state). Draftees under this section are again only for the non-officer ranks of the regular armed forces. The option is retained for Russian citizens without other allegiances to volunteer, in which case they might become officers or serve in the broader military/intelligence establishment, or for non-Russians to volunteer for the regular forces.
Russia's original law was fairly generous in exempting holders of other nationalities, as most countries with mandatory military service or registration do not make the distinction. For example, Israel and the USA are two countries where holding additional nationality is not a barrier. (Though in both of those cases, there are other exemptions which might be in play. The USA is also not currently operating actual conscription, just a registration system.) And there are plenty of countries without a draft at all.
In the context of the overall pipeline from "you are a regular person minding your own business" to "you are in uniform pointing your weapon at your country's designated enemy", militaries also have several opportunities to discard draftees or volunteers who they genuinely don't want, for whatever reason. It may be that even a country which obliges dual citizens to register and attend an assessment, would then choose not to make them train and fight. Or they may be channeled into different roles. It might depend on the conflict in question (e.g. you are a citizen of A and B, and A is at war with C, so you have to fight; if A were at war with B then your fate would be different).
Regarding
Why should they care whether someone also has citizenship of some other country?
they may be anxious about you having divided loyalties, or about angering the other country. These concerns would be more acute if the country is engaged in an actual conflict.
There are several treaties that allow multiple citizens to avoid having to serve multiple terms of military service in each country. The details depend on the countries involved, but the general idea is that if you are (for example) both Austrian and Danish, and live in Denmark, then you do your military service there and Austria does not mind. In that case, it's helpful that neither of those countries is at war, and certainly not with one another. What you cannot generally do is use both of your nationalities to cancel each other out, and avoid military service altogether. Equally, if you are Austrian and British (the UK has no conscription) then the government of Austria still wants you to do military service on their terms.
