Luckily for you, there was a comprehensive census taken of the Russian Empire in 1897 (only complete one ever taken of the Russian Empire). This census included Ukraine. Here it is if you can read Russian.

Now for the history of Ukraine. Ukraine, as you think of it now, was largely (though not entirely by current'ish' borders) an administrative division under the Grand Dutchy of Lithuania until 1569. From there it was integrated into the Polish-Lithuanian Commonwealth. Of note is the Kiev Voivodeship was the largest stakeholder in terms of land and population but there were other divisions as well.

From the PLC, most of what is now Ukraine fell under the Russian Empire through the Partitions of Poland. There were many Governorates, or administrative divisions, that are "Ukrainian" based on language and culture but were member states of the Russian Empire.

Since your family said they're Ukrainian but officially listed themselves as Russian, they were identifying as Ukrainian based on the Governorate, or administrative region, they were from. Since it wasn't a country, they defaulted to the nation/country/empire whose borders they fell under.

Ukraine existed briefly as a country (not official, but it styled itself autonomous) following the October Revolution but were defeated by the Bolsheviks and reintegrated into the USSR, again as an administrative region where it wasn't a country again until the fall of the Soviet Union.

EDIT: Added dates and additional information for clarity.