Similarity between personal pronouns in French and Russian

Yes, the similarity is largely due to descendance from a common language (the reconstructed Proto-Indo-European). English too developed from this language, as did a bunch of others.

I'll limit myself to a few case forms and two numbers (PIE also had a dual) to keep this simple.

1st Person

The nominative singular is reconstructed as *eǵoH or similar. Capital H is some kind of back of the mouth fricative (like the French R or Х in Russian or Г in Southern Russian or in Ukranian) that was lost in (almost) every descendant languages but changed the vowel next to it in a way that allows us to reconstruct it. The accent mark on the ǵ indicates it was soft, like Гь.

In the ancestor of Russian and the other Slavic languages the soft G turned into /z/ and the first vowel became diphthong, leaving us with something like *(j)azъ /'jazu/. This eventually shortens to Russian я /'ja/

In Latin, the soft g became hard and the pronoun was egō. In vulgar Latin, the /g/ was weakened then lost entirely, leaving just eō /'ɛo/. In the development of French, stress shifted to the second vowel and the first vowel turned into /j/ then into the modern /ʒ/. The word only survives in its unstressed form "je" /ʒə/, but there was an old stressed for jo /(d)ʒu/ used in the Middle Ages.

Whatever similarities still exist between the two pronouns are mostly coincidental, as you can see.

The accusative singular was *Hme (I'm ignoring the difference between the different kind of H for brevity here). The H was lost without impact in both the lines leading to French and to Russian

In the ancestor of all Slavic languages, the pronoun blended with a genitive form *mene, to become мене /mɛnɛ/ and eventually меня́ /mʲe'nʲa/ in Russian.

In Latin, it remained mē (with a long vowel), which eventually became both me (when unstressed) and moi (when stressed) in French. The orthographic similarity between moi and мой is a coincidence (besides both starting with /m/)

The nominative plural was *wei. This is obviously continued by English we but not so much by Russian мы /mɨ/ or French nous.

In the case of Russian, the initial consonant changed to /m/ for an unknown reason, possibly the influence of the singular in /m/.

In Latin, the nominative was remade based on the oblique cases.

The accusative plural was *nsmé (the n was the vowel of the first syllable). In both Italic and Slavic languages, this was shortened to the first syllable only.

In old East Slavic, this turns into нꙑ /nɨ/ but was already being replaced by the genitive насъ /'nasu/, the ancestor of Russian нас /nas/.

In Latin, ns became nōs (both nominative and accusative cases) which became French nous.

Here the similarity is in fact largely due to common descend.

2nd Person

The nominative singular was *tuH. The evolution to Russian ты /tɨ/ and French tu is straightforward.

The accusative singular was *twe. This gives Latin tē, and eventually te and toi, under the same circumstances as the me and moi.

In Old East Slavic, it again blended with another case form to give тебе /tɛbɛ/ and eventually Russian тебя /tʲe'bʲa/. Here the comparison should be to the Latin dative: tibi (no survivor in French, but in my other native language Walloon it became the stressed pronoun ti)

The nominative plural was *yuH(s). This again is obviously the source of the English "you", but not so much of French "vous" or Russian "вы" /vɨ/.

Both come from the oblique stem, which has a variety of reconstruction but is generally agreed to have started by /w/ (a sound which turned into /v/ in both Russian and French). The rest is pretty similar to the first person (and both pronouns influenced each other)

3rd person

PIE had no real 3rd person pronouns, but it had a number of demonstratives which eventually became its descendant's 3rd person pronouns.

The Russian ones come from two demonstratives: *Hen which doesn't really have a counterpart in Latin or in French (for forms like он /on/), and is/yos, which existed in Latin as a kind of equivalent of modern French ça (is, ea, id) but hasn't survived at all in Modern French (for forms like его́ /je'vo/)

The French ones come *Holnos > *olnos > Old Latin olle (with a suffix change) > ille. Le, la, les, lui, il, elle, eux and leur all descend from distinct case forms of it. The only possible equivalent in Russian is лони /loni/ (from an older olni) meaning last year (from an older "that year", retaining the same far demonstrative sense that "ille" once had)

Where we again see some similarity is in the reflexive pronouns (French se, soi, Russian себя /sʲe'bja/) which both descend from PIE *swe (with the same stressed/unstressed changes in French as the other persons, and the same replacement of the accusative in Russian. Once again, the Latin dative "sibi" is even closer to the modern Russian). The same pronoun also serve as the source of the possessives in both languages (suum > son/sien and свой /svoj/ for example)

So there you have it. There is some similarity, especially in the initial consonants and it's usually because of a shared inheritance, but it's not exactly systematic.