Here in the US, the words began to be commonly taught in the early grades during the "New Math" era. I remember in first and second grade (ca. 1972) being taught this distinction and thinking that it was dumb. In this context, kids were taught that "cardinal" numbers were for counting how many things you had, while "ordinal" numbers were numbers like "first," "second," etc., which told you an item's position in a list. This may have mostly died out along with other goofy and ill-considered features of New Math, but if you google, you can still find materials online such as worksheets where kids are taught the two terms and asked to match up 3 with "third," etc.
And of course if some piece of silliness like this makes it into the lower-grade curriculum, it becomes something that has to be taught to preservice elemetary teachers as part of their education coursework. They don't have any idea of the context where this distinction would actually be useful, with infinite sets and so on. They just parrot it like all the other crap they are forced to parrot in the classes taught by a university's education department. So if your question is when this distinction is learned and when it's needed, one answer is that it may be learned when people are studying to get a teaching credential, and needed when they need to take a multiple-choice test that proves they know enough math to be elementary school teachers.
There may be a more general phenomenon here, which is the perceived need to make isomorphisms more explicit, even when nothing is gained thereby. For example, Euclid didn't need separate concepts of angle and measure-of-an-angle, but it seems to be standard these days in high school geometry books to treat these as separate concepts, and to introduce a postulate saying that there is a mapping from angles to real numbers, which has certain properties.