Соня is indeed a very common Russian name, but it’s the diminutive form of Софья (Sophia) from the Greek word for wisdom. I don’t know, but it wouldn’t seem unlikely that when choosing a diminutive form people noticed it was the same as the word for a cute little animal.
I don’t think Lisa and лиса have any connection. There doesn’t seem to be a common Russian name similar to лиса. The English name Lisa is just a shortening of Elizabeth (Elisabeth) that’s become so common that it stands on its own now. Elizabeth comes from a Hebrew word for “God is my oath.” The only etymology I can find for лиса says it comes from a Proto-Slavic word and doesn’t say anything about any Hebrew origins. I don’t think there is any connection, but you can’t rule out that any family of Slavic background who names their daughter Lisa is aware of the connection.
This citegives Liza and Lizaveta as Russian variants. I don’t know if they would count as common or rare, though.
There is only a finite number of sounds that human voices can produce. Sometimes, you are bound to get the same sounds cropping up in two languages meaning two different things, and with no etymological connection. An example is the French verb “to break wind” (peter).
Lisa (the English name) is /ˈliːsə/
Лисá (Russian for “fox”) is /lʲɪˈsa/
Different L-sounds, different vowels, different syllable stress. Total coincidence.
Сóня (Sonya) /ˈsonʲə/ does appear to be the same pronuncition for both meanings, and more similar to the English /sɔ.nja/ since the names actually related. I think it’s coincidence, but it helps that doormice are cute. “Doormouse” doesn’t appear to be used as a given name in other European languages from a perusal of translations here.