Nightblood consumes all three aspects actually, Physical, Cognitive and Spiritual (and it's why some people in-world believe Nightblood doesn't just kill a person but obliterates them, so they can't go to the Beyond), just like Shardblades cut on all three Realms. It's why a fully drawn Nightblood turns it's victims' bodies into black smoke.
The confusion in-world regarding which part is the soul: the Cognitive mind or the Spiritweb is part of the point, I think. The Cognitive part is what may become a Cognitive Shadow (but are they really the same person or Investiture copying that person?) or disappear/go into the Beyond, Spiritwebs linger on in the Spiritual Realm long after a person dies, as long as there is some Connection to them, it is them whose voices people in the Cosmere hear from beyond the grave.
Spiritwebs can be interacted with by various magics, from Spiritual Connection via the Surge of Adhesion available to Bondsmiths, via Hemalurgy, via Soulstamps, etc.
The Cognitive part can also be interacted with. We've gotten some hints towards that that's how mental powers like influencing emotions might work, though note this is not explicitly confirmed. Soulcasting also involves using the Cognitive aspect of people/objects. Magics from Yolen heavily made use of the Cognitive.
Since, storing Connection and Identity is possible, metalminds can store parts of the Spiritweb. Storing plain Investiture is also possible, but we don't know if you can just... store a person's entire Spiritweb or Cognitive aspect in a metalmind, seems kinda weird.