At Epcot, it's a little more than the passport. I believe there's always some kind of "craft" thing as well. Right now it's Duffy The Bear On A Stick. If you go to the kidcot tables in each country, they will use the passport stamp on the Bear's stick (as well as in the passport). Imagine a cardboard bear on a big popsicle stick. The tables had books showing how you could color the bear to dress him up like the clothing of that country. I think they had markers on the tables. So basically, if you don't want to spring for the passport, your child can still get the stick bear for free and get it stamped.
My nieces were determined to go to every country to get their stick bears and passports stamped, so we did it over two evenings. I thought it was a pain trying to find the kidcot tables. I suppose if you had planned to look at everything in World Showcase and were doing the kidcot tables as an afterthought, you would bump into each table eventually. But if you are making a point of just being there for the kidcot tables, you have to search them out in each country.
We didn't do it at Animal Kingdom, so I can't help you there.