Multiple cutscenes in 5.3 seem to suggest that Emet Selch was actually Azem's boyfriend. When revisiting old 5.0 cutscenes, it seems canon at this point that Emet Selch was actually Azem's lover/boyfriend. We don't know if Ascians marry but if they do it's also very possible that Emet Selch is Azem's husband.
1. Emet Selch engages in friendly banter with WoL/Azem
Emet Selch said this out of nowhere, which seem to imply that Emet Selch expects WoL/Azem to want to engage in friendly banter with him. It's pretty typical for very close friends and lovers to engage in friendly banter with each other. You typically would not engage in friendly banter with mere acquaintances or strangers. What's more it's exceedingly rare for WoL/Azem to actually banter with the Scions (there are at most two or three times when they teased a Scion). So for WoL/Azem to actually banter with Emet Selch in the past kind of hints at a deeper relationship than even what WoL has with the Scions.
2. Emet Selch and Azem have the same element
To quote this Reddit post, "Nald'Thal has some functions, but one especially is interesting, he is the keeper of the hell/underworld. Emet Selch is also named Hades, who is a reference to the greek mythology, and is linked to the hell/underworld." At the same time, both Nald'thal and Azeyma have the same element, and we know that Azeyma is just a distortion of Azem. This is another piece of evidence that Emet Selch might be Azem's boyfriend.
3. Emet Selch senses a hint of you
With the speculation that other scions could also be sundered Ascians (maybe part of the Twelve that summoned Hydaelyn), it is weird that Emet Selch only sensed WoL/Azem. It's probably because Emet Selch had a very strong bond with you. Hythlodaeus also implicitly hinted so, because he basically portrayed himself like a good friend to a couple (you and Emet Selch).
4. Emet Selch returned your identity to you
Hythlodaeus once again told you that you are Azem but most importantly hinted that Emet Selch was the one that gave you your constellation crystal. Although one reason is that he wanted you to help put Elidibus out of his misery, also implicit in this act is a deep trust in your sense of duty and justice and compassion. How would Emet Selch know that you are capable of this? He must have known you well before the sundering. And with him imploring you to remember him, it probably hints at a far deeper relationship than what a surficial reading of the story suggests.