It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the integers or the reals. A square is just the result of a binary (product) operation where both of the operands are equal. The operation can be defined on an algebraic object with much richer structure than the integers or reals, and the operation itself can be much more complex than, say, integer multiplication. So the geometric product of geometric algebra is just one animal in the zoo of examples. You might call the specifics of the geometric product operation an "invention," but not because the square of a non-number can be a number.

Well if one would consider numbers and their product "discovered", then my point was that introducing the objects which have the property that they are not numbers but square to numbers seems to me like an invention.

At least if there is to be any meaningful distinction between a discovery and an invention.

> If B follows from A via logic, but A is invented, isn't B invented as well then?

Not really, because you have no choice about B - it's true (or rather it follows from A) whether you want it to or not. You could have picked a different A, but once you picked A then B was fixed.

Yet if I pull the trigger we say that I killed someone, not that the bullet did.

People invented the electric chair because people discovered that high-voltage electricity can kill people. You might invent a particular axiom because it has an interesting mathematical consequence (e.g. the axiom of choice was invented to allow Hilbert's basis theorem to be proven), but what you invented was the axiom, not the consequence.