Most of the "lore" comes from talking to merchants, and the descriptions you get when mousing over the ship type in the hangar. The story, especially concerning the vault of souls and the commander, is really open ended and vague, in my opinion.
The developers are independent and not associated with Subset. They don't hide the FTL influence, and have posted several times in r/ftlgame advertising Void War (among other subreddits). If I had to speculate on why they don't make references to FTL in the marketing material and FAQs, it's probably because they want their game to stand on it's own, rather than being a derivative game "for fans of FTL."
One of the developers in the discord said "the marketing for this game has been kind of a nightmare [...] I kinda wonder if people just need some time to get used to the idea that something so directly inspired by ftl could actually be good. I'm pretty confident that quality will win in the end. Just gotta give it some time"