I came across a Reddit post that has some research into this problem...
Today was the first day I have seen Katrina in my town (since it may be relevant: I do not time travel). The only other time I've seen and gotten a fortune told from Katrina was once in a friend's town. With this in mind, I decided to see if I could make villagers and then demolish their houses over and over after getting their fortunes read and have it count toward the 20 person total. After having my mayor and my pattern holding character get their fortunes read, I made 17 villagers (same name, face type, and house location). This gave me a total of 19 people getting their fortune read in my town. I went ahead and went into her tent, talked to her, and tried to leave on my mayor character again and she requested the project! So from this, I can conclude that:
-Only players who live in your town can contribute to your total.
-It does not matter if you have your fortune read in your town or another town, it still counts for you.
-You can remake villagers and delete them to contribute to your total.
-Having people visit you does not contribute to your total, instead it actually counts towards your visitor's total 20.
-You can get Katrina to suggest the project on her first visit to your town if you fulfill the requirements that day.
I do not know if the fortune shop will count for sure, but I imagine it would. So you could have someone come to your town each day to get their fortune told and they will eventually unlock the shop in theory. I do not know if Katrina will suggest it to them at the shop or if they will need to wait until she comes to their town, however. I'll do some more testing with this to find out.
TL;DR - If you are paying people to visit your town when Katrina is visiting they should actually be paying you!
So in short, the redditor found that visiting Katrina in any town will increment this count for the town that the player visiting Katrina is from. In other words... if you are visiting Katrina in a friend's town, the counter for your town will increment, but the counter for your friend's town will not.