You might be surprised by how much can change in the world of labor and delivery from 1999 to 2018.
I was astonished by the differences in how things went with my first child as opposed to my last and they was just under a decade apart. And walking around right away was one of those changes. I was shocked when they came in to get me up and about because it was a lot earlier than with my first who I had close to the same year you mentioned. My first stay I had to be wheeled out of the hospital, but the last, we just walked out. I thought I was "in the know" about everything when I started that pregnancy, but things kept going differently than I expected based on my earlier one.
Length of stay can vary, too, especially if there were any issues with her. When I developed an infection with one of my kids I was given the option of staying inpatient or going back and forth to the hospital every day for a week of daily shots. We chose to come back each day, but if I had a baby still in the hospital, then for sure I would have gone with staying there instead.
Of course, it could also (and probably is!) simply be because the show doesn't always care about those kind of details, and as an example, i once again grumble about the lack of reality in the fire episode.