It only really matters when your finger touches a string that needs to be sounding, and stops that happening. Touching another string that isn't played is not a problem. And in fact, you'll find out later that touching some strings in that very way is actually a necessary skill to achieve!
It does take time and practice to get accurate fretting when one is a beginner, so just keep going, bearing in mind that often you will be pressing too hard, and that in itself will spread your finger onto adjacent strings.Examine which fingers you are fretting with, and if possible, use the thinner fingers.
It may be worth considering a different guitar, with a wider fingerboard, thus wider spaced strings. Maybe not what you want, but classical guitars usually have wider fretboards.
I've played for a long time, still do most day,(guitar and bass) and haven't had callouses for over 60 years. They're often the product of guitars that need a good set up, or not-so-good technique. Some players swear by them, they're welcome! Anyone reading this, please tell why you disagree!