In the picture are some fondant discs with "Happy Birthday", "mom", and single letters.

How do you think they make these?

I was thinking maybe a stamp, color the stamp with black icing or black edible marker, and then stamping onto the fondant.

You thought right, in my opinion. That's what I would do anyway. And what look like teapots in the background look to be done the same way. Others will tell you that the rubber used on stationary/craft stamps are not food safe necessarily, but each to their own

You can buy stamp pads that have no color yet. Then you can put your edible food color in that and use a rubber stamp.