Mathematics is very creative, in that you are allowed and encouraged to make things up from scratch. There are no restrictions about your creations matching the physical world (as in natural sciences), or being useful (as in engineering and technology). What you make up is not even required to have an aesthetic value; it just has to follow its own logic, which you are also welcome to make up. You can search the web for "algebras" or "geometries" (plural) to see pretty exotic stuff! But there are certain values and traditions within math as a human endeavor, such as consistency, precision, elegance, and so on.
You write, "Even though math seems to me to be closed into a world of rules and proper procedures it somehow allows itself to not pre-judge outcomes. I don’t know why that is." The seeming contradiction is resolved by viewing math as an open, creative world. It can be an allegory or analogy for something, but at its heart it's metaphoric. This brings me back to metaphors vs. analogies.
Both are two-parter structures with sources and targets. But analogies have pre-determined, pre-judged, pre-solved target. In contrast, metaphor is a tool for generating your own targets. I like the social links in your last paragraph (not seeing humans as individual processing units), because other people's suggestions, references, cultures, etc. mediate targets of our metaphors. But analogies not just mediate - they prescribe. Which is perfectly fine in many cases, for example, when bringing up a novel context. Say, to start playing with a non-commutative algebra, I might use the analogy with unrequited love. If I play with a toddler, I would use a sillier analogy, like the chair sitting on you not being the same as you sitting on the chair.
In terms of footprints, metaphor is a near-boundary tool, while analogy is a near-center tool.