This question gets really complicated really fast. After the breakup, each of the former Soviet republics established its own set of laws, and then these laws were rewritten multiple times. The region also includes half a dozen unrecognized states (Transnistria, Artsakh, Abkhazia, Ossetia, DPR, and LPR) each of them having its own very original definition of citizenship.
I'll try to give a general answer, but if you need to investigate a particular case you should probably ask a much more specific question.
Citizenship in the Soviet Union
USSR had uniform citizenship for all republics and a common passport for all its citizens.
Article 33. Uniform federal citizenship is established for the USSR. Every citizen of a Union Republic is a citizen of the USSR. The grounds and procedure for acquiring or forfeiting Council citizenship are defined by the Law on Citizenship of the USSR. When abroad, citizens of the USSR enjoy the protection and assistance of the Soviet state.
Consitution of Soviet Union, 1977
Republics had to respect the law of the Union and their constitutions usually reflected that:
Статья 31. В соответствии с установленным в СССР единым союзным гражданством каждый гражданин Казахской ССР является гражданином СССР. ... Граждане других союзных республик пользуются на территории Казахской ССР одинаковыми правами с гражданами Казахской ССР.
Article 31. According to uniform union citizenship established in the USSR, every citizen of Kazakh SSR is a citizen of USSR. ... Citizens of other union republics on the territory of Kazakh SSR have the same rights as the citizens of Kazakh SSR.
Constitution of Kazakh SSR, 1978
So, the citizenship of a republic was vaguely defined and didn't mean much. For example, one didn't have to apply for a new passport to migrate from Tajikistan to Lithuania. There were other bureaucratic hurdles, though.
Identity documents in the Soviet Union
The Soviet Union had a concept of residency permits (see, propiska). This means that Soviet people had to notify the authorities and receive an inscription in their passports to be allowed to move to another city, even if both cities are in the same republic. The requirements differed from city to city and could be very complicated at times. As a result, every Soviet citizen had a verified physical address written in their passport.
USSR also issued other types of IDs like birth certificates and employment record books (obligatory for all working people).
Also, the ethnicity (natsional'nost') was recorded in the passport. Some of the former republics still continue to record the ethnicity in official IDs and occasionally this can let someone fast-forward the immigration process.
Citizenship based on residency after the breakup
For the first couple of years after the breakup people continued to use Soviet IDs. It took a couple of years to implement a new legal framework.
Most of the newly independent countries accepted all permanent residents as citizens. To give an example from Central Asia, here's a law enacted in Kazakhstan ( On citizenship of the Republic of Kazakhstan, 1994):
3-бап. Қазақстан Республикасының азаматы болу
- осы Заң күшіне енгізілген күнге дейін Қазақстан Республикасында тұрақты тұратын;
- осы Заңға сәйкес Қазақстан Республикасының азаматтығын алған адамдар
Қазақстан Республикасының азаматтары болып табылады.
Rough unoffiical translation
Article 3. Belonging to the citizenship of the Republic of Kazakhstan
Those people are considered as citizens of Republic of Kazakhstan:
- people who permanently live in the Republic of Kazakhstan at the day this law is enacted;
- people who receive the citizenship according to the process established in this law.
So, the proof of residency (propiska) was enough to receive citizenship.
Soviet IDs were in limited use long after that. People were allowed to exchange Soviet passports until 1999 in Kazakhstan and 2004 in Kyrgyzstan (according to Wikipedia at least). I believe Soviet birth certificates can still be used in some exceptional cases (e.g. people who don't have any other kind of ID).
Ethnicity was never the requirement to receive citizenship, but many countries established a fast-forward naturalization process for people returning to their historical homeland.
Estonia and Latvia
As far as I can tell, of all 15 republics, only Estonia and Latvia refused to give citizenship automatically for all their residents. They only recognized people who lived there before the Soviet occupation in 1940.
Article 2. Possession of Latvia citizenship
Citizens of Latvia are:
- persons who were citizens of Latvia on June 17, 1940 and their descendants who have registered according to the procedures established by law, except persons who have become citizens (subjects) of another state after May 4, 1990;
- persons who have obtained the citizenship of Latvia through naturalization or another manner according to the procedures established by law;
- children found within the territory of Latvia whose parents are not known;
- children with no parents who live in an orphanage or boarding school in Latvia;
- children both of whose parents were citizens of Latvia on the day of birth of such children, regardless of the place of birth of such children.
Law of citizenship, Latvia, 1994
The rest of the residents had to go through a naturalization process to receive citizenship. The Law includes very detailed restrictions (Article 11, e.g. convicted criminals and former Soviet officials are excluded) and regulations (Article 12, e.g. citizen need to have a command of the Latvian language) for naturalization.
All former republics, except Estonia and Latvia, granted citizenship to all permanent residents willing to receive it. The Soviet Union was tracking people, so the residency was easy to establish.
"Ethnicity" (natsional'nost') was also easy to establish because it was written in a passport. I'm putting quotes here because natsional'nost' is a very rough version of ethnicity, e.g. people with multiethnic background didn't have an option to put multiple ethnicities in their documents. Anyway, ethnicity was never a basis or requirement to receive citizenship in former Soviet states. In some cases, ethnicity can help fast forward the immigration process, though.
Persons of Russian descent with a connection to Central Asia needed to have proof of residence to be granted citizenship in Central Asia and/or Russia. With enough dedication, it wasn't and it still isn't hard to receive citizenship in both countries. Cases like these are not unheard of, even in countries where dual citizenship is illegal.