Perhaps it is my pride talking, so do correct me if I'm mistaken, but... Who else knows your heart half so well as I?

And the guy's dead wife is there. At the ghost DMV. The dead wife is a ghost and he's the ONLY ghost who knows he's a ghost. In the ghost DMV. You make friends with the guy's dead wife (while waiting in line) (at the ghost DMV) and he tells you that the lunatic immortal wizard who's making you defeat his seven evil bureaucratic processes while you die of magic cancer before you can even fight him directly is really just a silly sentimental fella.

Then in the next expansion you get to live through the dead wife flashbacks yourself. You get to meet the dead wife when he's alive in the past, outside of the DMV. And it turns out he works at the real DMV, in the past. And he has pink hair. And he and the past-self version of the insane immortal wizard (who sent you to the ghost DMV) pull off a sick prank to help you close the timeloop.

Endless list of favorite things in Final Fantasy XIV  [7/?]

The warrior of light around the corner:

Sure, the journal entry is named Emet-Selch. But the first thing written there is that his true name was Hades. You also see this when you describe him to the Minstrel for his extreme trial.

Additionally, the description of the trial alludes to this as well. As when we talk about those we have faced in the First. We talk and refer to him as Hades. Which is also written similarly to the journal. Both of which were described/written by the WoL.

"Hearken unto a requiem for a hero fallen. A man who lived a thousand thousand of our lives clinging desperately to faint hope, never shirking his sworn duty to his long-lost brethren. A man who stood proud and did avow his true name on the threshold of the battle that would see him fall to his rival—the light to quench his shadow. Borrowing liberally from the funereal rites of the Night's Blessed, the minstreling wanderer weaves an elegy in that hero's honor—the tragic-yet-triumphant tale of a man and a battle that ne'er shall be forgotten."

You can also see this in the quest dialogue and while we cannot know the exact words the WoL used (as it is your own intrepretation of it) it is still clear that the WoL didn't refer to Emet-Selch as Emet-Selch they call him Hades.

For the WoL, this is about honoring the man who held steadfast to his ideas. Who fought for his loved ones just as much as the WoL does. Not the Ascian Emet-Selch. To honor and remember Hades as he once lived.

There is however, the matter brought up by the Minstrel: Why did Emet-Selch reveal his name to the Wol?

We do have a simple meta reason why: Hades is a recurring Summon across the Final Fantasy games. Using the name Hades is just natural to do so.

However, let's look at this from an in-lore perspective as well. For which we can look to what he says and speculate.

In the quest, Return to Eulmore, before leaving to Wright you can question Emet-Selch over the information he gives in the cutscene before. Revealing to us that Emet-Selch, along with the rest of the ascians encountered, is merely a title inherited. Their true names are hidden to take up the name and position of their seat.

You can, upon hearing this, ask him for his true name:

His reply to this is rather interesting:

There are a couple of things to note from his response. Firstly, he doesn't outright dismiss telling you his name, only says that eventually he'll reveal it. Of course, this hinges upon you living through your trials in putting down the Lightwardens and containing the light within, or simply dying from other matters.

But this would be disappointing for him. This dialogue ties into what he proposes to you later in The View From Above. To stand with him as allies. He doesn't propose this to the rest of the Scions, just the WoL. He dangles these threads because he wants them to reach back as Azem would. The WoL dying would be disappointing, and he would have to begin his search anew for Azem's soul.

We don't know if Emet-Selch has encountered Azem's shards before the WoL. Maybe he had or maybe he didn't. But it wouldn't change the fact that the WoL's death would have him searching again.

Even as he hurls insults upon the WoL for once more disappointing him, that is still Azem's soul in there. After all, his invitation to seek him out in the Tempest allows you to die with dignity. Everything he ever does is not let himself be alone and reach out to an old friend.

He wants someone else to remember it all. Who is more worthy of remembering it all than Azem?

Emet-Selch is a man of many masks. It is true, and his emotions are ever cloaked, but there are ever glimpses of them throughout Shadowbringers. Especially if it's Azem's soul prodding at him to reveal the layers underneath.

So in his final confrontation, when either the WoL dies or he, wouldn't it not be disappointing to leave the question of who the man underneath is all truly is? Perhaps even this even the last-ditch attempt to have the bearer of Azem's soul remember before either of you dies.