D&D 5e has some core rules-under-the-rules. The first is: The rules do what they say they do.
Given that, let's look at the rules text of this specific power.
While projecting your spirit, you gain the
• Your spirit and body gain resistance to
bludgeoning, piercing, and slashing damage.
• When you cast a spell of the conjuration or
necromancy school, the spell doesn’t require
verbal, somatic, or material components that
lack a gold cost.
• You have a flying speed equal to your walking
speed and can hover. You can move through
creatures and objects as if they were difficult
terrain, but you take 1d10 force damage if you
end your turn inside a creature or an object.
• While you are using your Form of Dread, once
during each of your turns when you deal
necrotic damage to a creature, you regain hit
points equal to half the amount of necrotic
So what does this tell us? Well, both the body and the spirit are subject to damage, as both receive damage resistance. By inference, both can be effectively targeted. Nothing is said about funny things happening with HP, so presumably it's all coming out of the same HP pool. Nothing is said about effects, so, by default, anything that hit either the person or the spirit would affect the whole. So... basically, they're both a creature, but they're both the same creature, who now happens to have two locations they can be targeted from. They can fly and pass through items (because it says they can) but they're not invisible (because it would have said they were if they were, and it didn't.)
Some of that is starting to wander towards the realm of "reasonable supposition", though. After all, another one of the underlying principles is Rule 0: The rules do what the DM says they do.
Unlike many other tabletop games, 5e actually leaves a lot of the funny little edge cases undefined, and subject to DM interpretation. The core of the game is balanced enough that most forms of reasonable DM adjudication won't break things one way or the other. That's pretty clearly what's going on in this case. Some of the potential interactions are left unspecified, even when some sort of an effect might be reasonably expected. (Does your astrally projected self keep the AC for your armor? Can he stab people?). It's up to the DM to make the call.
The final bit that's pertinent here is Unearthed Arcana Rules are for Playtest. On top of 5e's general inclination towards looseness at the edges and DM adjudication, the stuff that comes out of UA is not yet deeply analyzed or fully playtested. Testing this stuff out and figuring out the issues is what UA is for. Thus, it's often going to leave some of the implications undefined, and it's going to require even more DM adjudication than normal.
...and that finally rolls around to a bit of a frame challenge... because there isn't a grand unified "how do souls work" in 5e, unless your DM comes up with one. Effects that muck with the soul are few enough and far enough between that they'll each have their own specialist rules, which may or may not be entirely in alignment, but almost certainly won't clearly define everything. Basically, for any one spell or effect, if you want to know how it treats souls, then you need to read it (and read the entire thing - there is no crunch/fluff divide anymore) and if it's still not clear, you need to get your DM to adjudicate (or, if you're the DM, you need to make the call yourself).