Our star orbits around a galaxy, which you can see if you look up at night: The Milky Way. As you can see, it's not bright enough to really call a sun.
The problem is basically luminance density, or you can say candelas per cubic meter, or total flux over the smallest blob that envelops the whole galaxy. It needs to be a lot higher for what you say.
Making stars a lot brighter doesn't make much sense. However, you could pack them tighter. Most galaxies are not that packed, so yours would be unusual and perhaps you may need to explain that, but hey, it's science fiction.
Nothing obviously bad or weird would happen if the stars were a bit closer to each other.
One estimate would be to copy paste 20 solar systems, all at 40 AU from the Sun (orbits of Uranus touching). That gives you 20/1600=1/80 the light of the Sun from the first shell. The second shell, at 80 AU, would have 4x as many, so you get 80/6400=1/80. Shell 80 would be 3200 AU away, would have 6400*20=128000 stars, so the whole system is about 250k. For comparison, our galaxy has ~250 billion, a million times more. Of course, if you pack solar systems that close, something bad or weird probably would happen - the Uranii might collide, the Oort clouds, the orbits of the Suns... But you could space them out more and compensate by having more stars in the whole galaxy, so it seems plausible. Keep in mind that in this oversimplified calculation, the entire galaxy is only 0.1 light year across, which is odd, but it doesn't seem impossible.
Another way is to take the Milky Way and compress it. How much would you need to compress it, to give as much light as the Sun? Taking the galaxy at -5 and the Sun at -26, we it needs to be about 10^8 times brighter. That means things need to be 10^4 times closer. It's about 87k light years across, and instead would need to be 8.7 light years. Again, certainly odd, but perhaps not obviously impossible, and luckily "sort of" close to the other estimate.
One problem I see is that while the Sun makes a lot of nice visible light, a lot of objects in the galaxy make all sorts nasty stuff like X-rays. By compressing them and making them 10^8 stronger, you're also increasing the galactic radiation by a similar degree. That doesn't sound like good news for the people living there.