What really are the parallels between your game and Dark Sols? Or is it just because of the name?
It's really hard to picture how would Dark Souls design influence a shmup. Considering the answer is not just "difficulty" because then it could be called "Battletods x Gradius" without changing the game, lol.
I've actually been wondering that myself, lol. This will be a lengthy ramble...
One thing I've realized upon thinking about it more is that they are, to a degree, both in the same genre from a design perspective. If we compare the two, what do they have in common? Both are difficult, but as you said that's a comparison that can be made with any number of games. But they're also very similar from a gameplay perspective in that both games are about memorizing enemy placement and patterns, ditto for bosses, farming some form of points to upgrade the player, and having a high penalty for death. The player gets better in both games by playing and learning, and I think it's actually what made Dark Souls so popular in the first place. They bring back that feel of games from the 80s, where the player learns how to manipulate the game to better their advances. (So far, Battletoads could still work as a comparison here.)
I own the first and second Dark Souls, and I'm terrible at them. I haven't gotten very far in either since they're such a time commitment, but I enjoy them for the challenge they present. When I recently went back to Gradius on the NES at my parents', I was surprised to feel something familiar to when I was playing Dark Souls. It was a lot easier, mind you, but it still got me thinking.
That was a bit of a tangent, but one I was going to post about later but then this came up lol. They both feel similarly in design, but what I want to do with Distant Sol is I want to add a bit more
soul (sorry) into the game. I'm planning on doing this by adding things not found in Gradius, like an unconventional life system, branching paths, and a sparse but lore-filled story. That's all on top of the usual memorizing enemy patterns and approaches as well as unique boss fights.
I'm still figuring out a way to approach the life system, but the basics are that you start with one life bar at the bottom of the screen, and each enemy drops "souls" which you can collect to fill the life bar at the bottom. This is similar to Gradius where you could collect a power-up after killing a full pattern of enemies. After killing ten enemies and collecting their souls, you gain an extra bar (or life) at the bottom of the screen. Dying loses a bar, obviously, and losing all of them is a game over. But, if you complete a level with say, 5 bars, then you can spend them in a shop at the end of each level to give your ship a permanent upgrade. This does cut into lives, making it a bit of a risk/reward element. This part is kinda like the bonfire in Dark Souls, where you get a respite from the action to give you a better chance of survival.
Hopefully, that gives a bit more information why it isn't just "Any-Random-Game X Gradius".

I'm not trying to market this as a Dark Souls or anything, it just helps me keep my vision on what I'd like it to be.