There is always some special arrangement of the neurons that makes the difference between a brain and a mass of neurons. We can even say we associate words to shapes in an innate (culturally independent) way, as the Bouba/kiki effect demonstrates. But from that substrate there are many things that we learn.
To measure how much we learn we can take a look at children, as in the answer from cartomancer, or we can consider people that have different culture, capabilities, etc.
For instance we can consider blind people. Blind people are in general are more used to a three-dimensional world where two dimensional objects do not make so much sense. [They are still normal for them (more than 4D objects) as a wire can take any of those shapes, for instance.] The point is that our experience and perception shapes our understanding and comparing with haptic perception helps to understand where do our prototypes and concepts for shapes come from.
There are many notable blind mathematicians specialized in geometry allegedly due to this difference in perception of the world, shapes and geometry, that at the same time influences a difference in the understanding of geometry. I would personally say that a posteriori influence is what makes a difference in the a priori substrate.
To finish up I'd like to call the attention over a specific paragraph in the original question:
This means that although there are such geometries, because as human
beings we have only our immediate environment to purvey, that is, our
spatial knowledge is local, what is a straight line or circle in the
standard sense remains effective. It does not have to be acquired, but
can be innate.
The point about innate as being more effective does not make much sense. The same can be said about colors, for instance, however colors mean nothing to blind people and are perceived in different ways depending on color-blindness. That point suggests an intelligent design that creates people in ways that are efficient, however that is not the way evolution works and to the best of our knowledge that [efficiency] is not a good reason to think humans are in one way or another. We should be very careful about assumptions that are introduced inadvertently in such ways.
PD: Actually evolution pushes human beings to have the least amount of innate knowledge, the brain is not mature at the moment of birth due to limitations to enable birth in a bipedal species like humans.
Also, it may be interesting to consider spiders, and probably other animals. Spiders can make webs that can be perceived as complex to the human eye. We can debate about whether that is innate knowledge or knowledge at all. Spiders sure don't have an explicit knowledge that they can transfer to any other being, or reason about. Probably it is just a feeling, about what feels right at a given time making a web, and that feeling is altered by drugs. In this case I'd say there is a substrate and an emerging pattern through some kind of "spider feelings", but the spider never knew anything in an explicit way.