They seem to come from the same proto-indo-european word "weyk", which has two separate meanings. Is this a coincidence, or are the words related?
Apparently not, the etymology of victim is less clear.
mid-14c., from Anglo-French, Old French victor "conqueror," and directly from Latin victorem (nominative victor) "a conqueror," agent noun from past participle stem of vincere "to conquer, overcome, defeat," from nasalized form of PIE root weik- (3) "to fight, conquer."
weik- (3) Proto-Indo-European root meaning "to fight, conquer."
late 15c., "living creature killed and offered as a sacrifice to a deity or supernatural power, or in the performance of a religious rite;" from Latin victima "sacrificial animal; person or animal killed as a sacrifice," a word of uncertain origin. Perhaps related to vicis "turn, occasion" (as in vicarious), if the notion is an "exchange" with the gods. Perhaps distantly connected to Old English wig "idol," Gothic weihs "holy," German weihen "consecrate" (compare Weihnachten "Christmas") on notion of "a consecrated animal."
weik- (2) also *weig-, Proto-Indo-European root meaning "to bend, to wind."
I have said a few words about the latin words victor and victima. The old dictionary Lewis and Short are not helpful about etymology. 'Victor' is traced to 'Vinco', as cannot be doubted. For victima, I suggested that the verb vincio (I bind, as might happen to a defeated prisoner - they were brought into Rome in chains - vincula), with its supine vincitum or vinctum. Lewis and Short suggest vigeo, meaning I thrive as the root of victima.
"From Proto-Italic *wegēō (with unexpected i), from Proto-Indo-European *weǵ-eh₁-(ye)-, stative verb from *weǵ- (“to be lively”), same ultimate source of English English wake. Confer with the causative vegeō." Wictionary entry for vigeo
However, before we go further, there is an important point to be explained. It is the letter 'v' in ancient Latin. The shape that we call 'vee' (and the French call 'vée (vay), was pronounced like the letter u (roughly as in 'put'). Also, it was mostly seen as what we now think of as the capital V. So ancient Latin had no separate vowel u. That is because their V was their u. They had no vee sound.
BUT: when when the Latin 'V' is followed by 'a', as in 'validus' (strong) or 'o', as in 'vox' (voice), or 'vita' (life), it sounds like a 'W'. You can try it yourself: "u-eeta" turns to uweeta --> weeta. Interestingly, the same word with the same spelling,wagon. So, is pronounced with a double-U in English but a vee in French. So, Tmlen, the initial w in "weyk" is at least a plausible similar feature. I am no etymologist, but "weyk" reminds me of "wake" and "awake", and from there to life is not that much of a stretch.
Emma Dash suggests that the Latin noun victvs (that is how it was spelled) means life. In a way you could say that, but not quite. The noun is derived from a third verb: not vinco or vincio but vivo (I live). It is a fourth declension noun derived from the supine, and means nourishment, diet, even way of life.
The more I think about this, the more tempting to think that there might be a connection between these various words (victor, victus, vinctus, possibly even vita). But etymology is a colossal subject, and, as the poet and critic Samuel Johnson puts it,
A little learning is a dangerous thing.
Drink deep or taste not the Pierian spring.