Why does Mazikeen so desperately want to get that thing?

Question

Why does Mazikeen from Lucifer so desperately want to get a soul?

She is clever, intelligent, inquisitive. She is emotional and understands emotions in others. She adapts and thinks ahead. She has friends.

What does she expect a "soul" will bring her? Turn her into a mortal human?

Better understanding of human nature? Of human interactions? No, a soul does not bring this (whatever that soul is supposed to be). We have plenty of people that do not get these social contexts and they are not soulless.

Because she would not be "just a daemon" anymore? That would mean that ripping someone's soul turns them into a daemon. But this is not how she was made a daemon in the first place.

So what does she need a soul for?

Answer 1

The biggest thing would be immortality. People without souls don't go to Heaven or Hell. Should she die she would cease to exist. A soul in this instance is sort of an insurance policy.

The lack of soul also hits her on an emotional level because of her relationship with her mother Lilith. Because Lilith isn't God, she was unable to create souls. But she decided to create demons anyway which Mazikeen seems to resent. She feels incomplete because her mother basically created her to be a torturer in hell and have no other purpose. Now that Mazikeen is no longer in Hell, she has no direction or reason to exist. She may feel in this sense, that a soul would make her "whole" or "complete".

And finally a current theory, as in this is currently not verified in the show but it might in Season 5 part 2, is that Mazikeen has already developed a soul but is not currently aware of it. Prior to the events of the show and in early seasons, she was pretty "soulless". She didn't care about anybody. She was pretty self-centered with the exception of her undying loyalty to Lucifer. However, now she exhibits all the traits that you described in the OP. Her new found emotions may be the reason she suddenly cares about whether or not she has a soul.

EDIT:

The show has finished and there is a big spoiler in Season 5 Part 2 below.

It has been confirmed that Mazikeen did, in fact, grow a soul (or at least God implied she did, she jumped to that conclusion, and they never said it was wrong.)

Answer 2

She is looking for a Soulmate

All of Mazekiin's relationships with people rely deep down on either sex or violence: The important people in her life are either those that she wants to protect, those that she wants to hurt and those that she wants to f*ck. Sometimes all three at once. But when she notices the relationship that other people have... she notices that she is missing something. She tries to get it, but she can't, hence she assumes that there is something wrong with her.

Finally, after her slap-slap-kiss fight with Amanadel she comes to the conclusion:

[Maze]: You don't think I'm scary?

[Amanadiel] Oh, I think you are hella scary. But... it's good scary, because it's you. And I know when you finally show someone how absolutely amazing you are, you're going to find your soulmate.

[Maze]You can't find a soulmate if you don't have a soul.

I don't mean that she is right, but that she believes in it - in fact, she already has at least one great non-sexual friend in the form of Linda, but it seems that either she wants more or that she can't see it.