Why are there are typically 13 steps to every staircase?

Ergonomically and for safety reasons, stairs have to have certain measurements in order for people to comfortably use them.

Maximum riser height, typically 8.25 inches (210 mm). Note that by specifying the maximum riser height and minimum tread length, a maximum slope is established. Residential building codes will typically allow for steeper stairs than public building codes.

All that remains is basic math with regards to ceiling height.

https://en.wikipedia.org/wiki/Staircase

Because in most buildings, the second story is the same distance to the first. Or just simply zbeck's response.

I've got 14 between the ground floor and the first floor, not counting the floors themselves. (No second floor.)

In my building, there are 17 steps between each floor. Last place I lived there were 15. I was unaware that there was a standard anywhere for such a thing.