Question about a property of a plane?

Question

This is from Kiselev's Geometry: Planimetry, page 2:

One can impose a plane on itself or any other plane in a way that takes one given point to any other given point, and this can also be done after flipping the plane upside down.

I'm not sure I really understand what this means. My interpretation is that at any time we can make a copy of any plane, make it coincide with the original plane (or make it parallel). Then we can take take any point in the copy and shift the copy so that that point lies on top of any other point in the original. But if we are talking about infinite planes with no axes, I'm not even sure how much sense it makes to talk about individual points; once the shifting is done, the final situation seems indistinguishable from the initial situation (unless we perhaps color some points, but this has not been mentioned).

Answer 1

Well, it literally seems to say this

There is a bijection between any two planes.

However, I wonder if it doesn't intend to say something richer like

There is a bijective affine transformation between any two planes

or

There is a distance preserving map between any two planes.

But this latter one at least requires a notion of distance, and I don't know the context of the book. Since the language is geometric ("flipping the plane") I suppose that the author is probably working with an ordered field, and that would make the last statement useful.