How does an enlightened mind distinguish itself from all other phenomena? How does an enlightened mind distinguish itself from all other phenomena?
There is one such person amidst a crowd of other people. Somebody shouts their name, and they say, &quot;yes, that is I. I am here. What do you want?&quot;
After hearing his name, how does he make the connection between the hearing of his name and referencing that name to himself, so that he then responds thus, &quot;yes, that is I. I am here. What do you want?&quot;
The mind of a fully enlightened one is completely empty of discursive thoughts. It does not conceptualize at all. In fact, this has already occurred to an Arya being at the 7th Bhumi let alone a fully enlightened Buddha. So the answer to your question is that an enlightened mind DOES NOT distinguish itself from all other phenomena.
For references I draw from this commentary by Dzongsar Jamyang Khyentse Rinpoche on Chandrakirti's Madhyamakavatara.
Here is a screenshot and highlight of the table of contents of the pertinent sections:

When we turn to page 369 we see a verse describing the way in which an Arya being achieving the 11th Bhumi (full Buddhahood) knows suchness and associated commentary:

Right after this passage the next verse is an objection. I think the objection is coming from someone espousing the Cittamatra tenet system as this is the last labeled objector in the text. Anyway, here is the objection:

A little further down we see a verse by Chandrakirti himself giving a simile of how it can be that a Buddha's mind can be totally free from discursive thought all the while being of benefit to other sentient beings. This seems to be the thrust of the OP's question if I'm not mistaken:

Here's the really dispositive part. A little bit later in the text Rinpoche affirms that the standard interpretation of the text and of Tibetan Buddhist schools is that a Fully Enlightened Buddha is entirely free of discursive thought and elaborates a bit. He says the answer to your question is basically that an enlightened mind DOES NOT distinguish itself from all other phenomena:

There is so much more in this commentary and I encourage you to read all of it as it is an absolute treasure. Of course, this commentary and root text are from the masters of the Prasangika Madhyamaka and so keep in mind this as the basis for understanding the text.