Like I said, the galaxy is so inconceivably huge that naming something after the entire thing seems way too vague. It would seem more sensible to name an interstellar power after the region within the galaxy that it occupies. The problem is that it's hard to break the galaxy down into meaningful regions. Star Trek's four-quadrant system is arbitrary (using the plane connecting the galactic center and Earth as the prime meridian, as it were) and doesn't really reduce the hugeness by even so much as an order of magnitude; it's like naming your city on Earth after what hemisphere it's in. You could go with galactic arms, but those are also pretty huge and they wrap around the galaxy, so they aren't very specific either. Even just the small Orion Arm that we occupy is rather gigantic; if it were the size of a banana, then the entirety of Star Trek's 24th-century Federation would be the size of a gnat. (Okay, maybe a large gnat.)
There are some named regions in space that are large enough to encompass hundreds or thousands of star systems but small enough on a galactic scale to be reasonably specific, but they aren't very well-known. You can find some good star maps of the galaxy on various scales on the Space Maps page at Atomic Rockets (a site that has a lot of excellent worldbuilding and science resources for SF writers). Some of them are a couple of decades out of date, though. The maps from Henbest & Couper's The Guide to the Galaxy are excellent for giving a sense of the geography of galactic structures like star-formation zones, gas bubbles, and nebulae, but their star positions are based on pre-HIPPARCOS parallax data and thus are often erroneous.
There are some named regions in space that are large enough to encompass hundreds or thousands of star systems but small enough on a galactic scale to be reasonably specific, but they aren't very well-known. You can find some good star maps of the galaxy on various scales on the Space Maps page at Atomic Rockets (a site that has a lot of excellent worldbuilding and science resources for SF writers). Some of them are a couple of decades out of date, though. The maps from Henbest & Couper's The Guide to the Galaxy are excellent for giving a sense of the geography of galactic structures like star-formation zones, gas bubbles, and nebulae, but their star positions are based on pre-HIPPARCOS parallax data and thus are often erroneous.
