I studied Russian, and I've never heard anything like this. I also had a very quick google and can't find anything about it. I can find no evidence for this statement, in short.
Names like
Yekaterina and
Anastasiya,
Nikolai and
Ivan were and are used by people from all walks of life.
Maybe he meant that the serfs had more Slavic or ethnic (Tartar, Bashkir, Chechen, Chuvash,
Mari etc) names, or more often went by informal diminutive forms of address, eg,
Masha rather than
Marya, Dasha for
Darya,
Nastya for
Anastasiya, etc. For a while the upper classes were a bit obsessed with Western/French culture and wanted to be more French than Slavic, so that might have influence name choices too, or at least what they styled themselves as.
And maybe serfs would naturally pick less 'elaborate' names, but I can promise you that
Anastasiya and
Nikolai and
Yekaterina don't have and never really had the whole OTT 'regal' vibe in Russia that they do here.
Other than that, I've never heard anything like it, and would be hesitant to believe it...
I will say that there are two or three names I've heard being referred to as old-fashioned housemaid/servant names -
Anfisa,
Polina and
Faina.