"Does もの here refer to a voice?"
Yes, it does.
If so, would there be any difference between the above and the following?
There is no difference in meaning, but as you stated, repeating the same word only a few syllables after using it once like in this sentence would be avoided at all cost in serious writing. (I take your first sentence to be from a novel or something because it uses the historical present at the end in 「気づく」.)
In informal daily conversations, however, native speakers do not frown upon repetitions like this at all in most cases.
" would it not be also correct to omit the noun entirely and say:
This sentence is grammatical and even natural-sounding, but again, it would sound too conversational with 「誰のか」.
The best sentence for use in a novel is the original one.