According to the Russian Wikipedia, the Cyrillic spelling of the poet's surname is Есенин:
https://ru.wikipedia.org/wiki/%D0%95%D1%81%D0%B5%D0%BD%D0%B8%D0%BD,_%D0%A1%D0%B5%D1%80%D0%B3%D0%B5%D0%B9_%D0%90%D0%BB%D0%B5%D0%BA%D1%81%D0%B0%D0%BD%D0%B4%D1%80%D0%BE%D0%B2%D0%B8%D1%87 (in Russian)
This indicates that the Cyrillic spelling of
Yesenia is Есения, of which the proper transcription is actually
Yeseniya.
A Russian name website gives 5 possible meanings and origins for Есения, which includes the poet's surname as well as the genus of South American palm trees (as mentioned in BtN's entry for
Yesenia). In the case of the latter, the website mentions that the name became very popular in 1970 because of the series (yes, it specifically says 'series' and not 'film'). You can see it for yourself here:
https://imya.com/name/6891 (in Russian; use Google Translate if necessary)
The same website does not have an entry for Есенин (Yesenin), the poet's surname. If Yesenin was ever used as a given name for Russian boys, then it must have been extremely rare (like most Soviet given names). It might be mentioned in Herwig Kraus' book
Sowjetrussische Vornamen: Ein Lexikon (2013) under the germanised spelling
Jessenin, but the relevant page is not available for free on Google Books:
https://books.google.de/books?id=3tXmBQAAQBAJ (in English and
German)
With that said: it does indeed look like the given name
Yeseniya first came into use in the 20th century, as the name is not mentioned on
Paul Wickenden's website
A Dictionary of Period Russian Names:
https://heraldry.sca.org/names/paul/ (in English)
Lastly, even when you use the Cyrillic spelling, it is probably going to be quite hard to find bearers older than 50 on social media like Facebook and its Russian equivalent VKontakte, since the name must have been rare before 1970/1. It is possible that most (if not all) of the few bearers who were born *before* that year are either deceased or too elderly and computer illiterate to be on social media (or perhaps even too poor to be able to afford Internet; remember that poverty is still very much a thing in Russia).
~ If so is it possible that the Russian name actually influenced the use of the name in Latin America?
Since the name was probably rare in Russia even in the Soviet era, I highly doubt it. How would a Soviet Russian name even become popular in Latin
America? For example: it doesn't seem like there was a hugely popular Russian film with that name, in the same way that the Mexican film was hugely popular in the Soviet Union and popularised the name Yesenia/Yeseniya.
~ Or did some residents of the old Soviet Union just wrongly assume that the name of the character in the Mexican film was derived from the name of the poet?
If the film was dubbed instead of subtitled when it was first released in the Soviet Union, and most of the character's names were russified to boot, then some people might possibly have made that mistake. But since
Yeseniya as a feminine form of Yesenin was probably rare before the film's release, the number of people to even connect the name to the poet like that must have been small. It looks like the name primarily became popular because of the Mexican film's titular character, and not because of a coincidental similarity to the surname of a once-popular Russian poet, who at the time had already been dead for 46 years.
Forchta in biuonga quamon ouer mi, in bethecoda mi thuisternussi.
In ic quad: "uuie sal geuan mi fetheron also duuon, in ic fliugon sal in raston sal?"