Source is not known, unless they know your voice
There is likely no awareness of the source of Awakened Mind telepathic communication.
Unfortunately, there really is no specific Rule to support this, but if you look at some of the wording of the ability, it gives you clues:
You don't need to share a language with the creature for it to understand your telepathic utterances, but the creature must be able to understand at least one language.
With vocal communication, you know the source because you can hear them, possibly know their voice, and can see who is speaking. With Awakened Mind, it is clearly not based on your own language (you don't need to share one) and therefore doesn't have a 'signature' (whether sound, visual, etc.) to associate it with a specific person.
It is possible that if you've communicated with someone(thing) before they may not recognize your voice. In fact, Jeremy Crawford did rule in favor of that as shown in J.A. Streich's answer to this question. If the target hasn't heard you speak, it's pretty clear they wouldn't know the source. If they have and do know you, then your option is to try to Deceive them via an ability contest. However, your DM may rule they don't remember your voice if it wasn't 'memorable.'
Voice of God - not Conversation with God
In addition, this is only for ONE-WAY COMMUNICATION. This was confirmed in the published Sage Advice 1.14
The feature is intended to provide one-way communication. The warlock can use the feature to speak telepathically to a creature, but the feature doesn’t give that creature the ability to telepathically reply. In contrast, the telepathy ability that some monsters have (MM, 9) does make two-way communication possible.
This limits the voice of god options a bit - they can't talk back. But you can still give commands and berate them for not doing it - you just can't have a telepathic conversation about it.