Digitally, it's more feasible. But if a star isn't generated until you visit it, then there is no connectivity between systems, and you can't really say your game / fiction is really about a whole galaxy. It's just about a series of disconnected star systems. And until you've done the generation then you don't even have that: you just have a funky little bitmap. That's like me showing you a picture of the Milky Way and saying that's a fleshed-out fictional setting.
But even then, with exactly two bytes per star system, your galaxy takes up over 186 gb of storage space. If you average four planets / features per system, one moon/feature per planet, and give each of those two bytes of description, you've inflated it to 1.67 tb of data. But you need little spacer bytes that say "move on to describing the next body" so between each needs to be a byte at zero.
Our solar system would look like this:
Single Medium Yellow Star
Tiny Rock Planet, No Moon
Small Gas Planet, No Moon
Small Rock Planet (Atmosphere), One Small Rock Moon
Loose Asteroid Belt
Small Rock Planet, Two Tiny Rock Moon
Huge Gas Planet, 62 Tiny Rock Moon
Large Gas Planet (Rings), 33 Tiny Rock Moon (1 Atmosphere)
Large Gas Planet, 27 Tiny Rock Moon
Large Gas Planet, 13 Tiny Rock Moon
If every system had that level of detail, you're looking at about 27.8 tb of data.
And I'd say to have a meaningful interaction with Earth you'd need to know more about it than that it is a small rocky planet with an atmosphere. If you accounted for space to tell what the main atmospheric composition was, the presence of liquid water, biology, civilization, not even getting into any names or specifics, you're still looking at a galaxy of well over 300 tb of data.
But even that doesn't actually give you a setting. You need to name civilizations, determine what civs they know about and how they feel about them, what their tech levels are, how peaceful they are, what atmospheres they breathe, etc. At that point you might as well call it 5-10 pb of data for your galaxy.
Don't even try for ship floorplans, names of countries or famous individuals, social movements, general maps of planets, and so forth.
Space is just too big to try talking about a galaxy.