I could spend two days typing a response. And might just do so at some time but I recently did a deep dive on Illithid lore to try and find out the answer to this question.

The briefest possible answer is that they do have souls but souls that can only

1. Join with an elder brain
2. Join with one of the Illithid gods
3. Join with one of the powers of the far realms

Your soul cannot be claimed the deity you worshiped before transform unless wish, miracle, divine intervention etc. So "soulless" works as a shorthand for cannot be claimed by Faerun's gods / can only be claimed by an evil entity.

This is the source that deals with it best:

https://forgottenrealms.fandom.com/wiki/The_Illithiad

Have you ever asked Gortash's corpse this question?
Bane answers and tells you it's a scorched earth campaign. All the other gods lose their followers which cause lose their power and the dead three are the only gods left standing.