For space operas there’s always Asimov’s Foundation series and Herbert’s Dune, the latter being much higher quality literature imo, but both with wonderful stories. I always pull planet names from those universes when I’m playing any human empire and name stations and fleets after characters. Neither have aliens, though, so they don't quite fit the feel of Stellaris entirely, so it's up to you whether or not a lack of filthy xenos interests you.
For military sci-fi Heinlein has plenty of good choices, with Starship Troopers being the obvious one. He has a military background, so it makes its way into a lot of his books. Forever War by Joe Haldeman is also very good and deals with some pretty crazy time dilation and the travails of interstellar warfare; it delves into issues that could arise from first contact and how a lack of information and the difficulty of long distance travel and communication can affect warfare and the soldiers involved in it.
Downbelow Station by C.J. Cherryh is a good read about interactions with primitives on a newly found/colonized planet that's not quite habitable for humans. She, of course, has a patch named after her, as do Asimov and Herbert. Speaking of which, any of the authors the patches are named after are obviously good choices.
For more modern space operas that I find to be not quite on the level of the classics, but still good, there's the Expanse series by Corey (pen name for 2 guys, I think), and the Red Rising series by Pierce Brown. Both take place within the Sol system (sshhh), and both are a bit more personal and character driven than the others, I think. I haven't seen the TV series for the Expanse, but I guess you could watch that if you don't feel like reading. Red Rising I would consider to be a less sophisticated and quicker read, but still pretty solid nonetheless.
As I'm writing this, I'm kind of realizing I really don't know of any Stellaris galaxy style books that involve a multitude of species in a mature and interconnected galaxy other than ones based on other forms of media (e.g. Star Wars, WH40K, Star Trek). While I enjoy some of those franchises, I've never really felt compelled to read into their expanded universes. If anyone knows of any original sci-fi books that are not based on those existing franchises that run in that vein, I'd also be interested in hearing any suggestions.
