Here is the question:
So did we discover symmetry, or do we impose it? (like numbers) And why do we need it?
David John Baker describes symmetries (page 2):
...symmetries of a theory are transformations that preserve its laws.
These transformations help us distinguish between "fundamental quantities" and "surplus structure" (page 4):
...physical quantities that change under symmetry transformations (i.e., that are not invariant) must not be fundamental quantities. Qualitatively identical objects or worlds cannot disagree about the fundamental quantities.
This provides a partial answer to why we need symmetries: they help identify the fundamental quantities of a physical theory.
They also help ground the idea of "objectivity". Brading, Catellani and Teh while surveying the philosophical significance of symmetry note:
It is widely agreed that there is a close connection between symmetry and objectivity, the starting point once again being provided by spacetime symmetries: the laws by means of which we describe the evolution of physical systems have an objective validity because they are the same for all observers.
A more complete answer to the question why we need symmetry would then be that the idea of transformations preserving the laws of the theory allow one to (1) identify the fundamental quantities of the theory and (2) describe the theory objectively, that is, independently of individual observers.
The other question whether we discover symmetry or impose it can be described from an ontological viewpoint or from an epistemological viewpoint.
Brading, Catellani and Teh describe the ontological viewpoint as follows:
According to an ontological viewpoint, symmetries are seen as a substantial part of the physical world: the symmetries of theories represent properties existing in nature, or characterize the structure of the physical world. It might be claimed that the ontological status of symmetries provides the reason for the methodological success of symmetries in physics.
The epistemological viewpoint gains strength when one doubts if symmetries can "be directly observed". From this epistemological perspective symmetries need only be approximate enough "such that there is sufficient stability and regularity in the events for the laws of nature to be discovered."
So, which one is most likely correct, the ontological viewpoint or the epistemological viewpoint?
Since science is provisional, that is, it doesn't claim to provide absolute answers but only those that are falsifiable, a safe position to take is the epistemological viewpoint.
Another reason to take the epistemological viewpoint is that symmetries also allow for objectivity. They abstract away the subjective part of reality to more conveniently work with a theory that applies to everyone. That makes the theory useful like taking a photograph or looking in the mirror are useful, but because of that they are only approximations for the reality of what was reflected in the mirror or imaged in the photograph.
Baker, D. J. (2010). Symmetry and the Metaphysics of Physics. Philosophy Compass, 5(12), 1157-1166. http://philsci-archive.pitt.edu/5435/1/SymPhysMeta.pdf
Brading, Katherine, Castellani, Elena and Teh, Nicholas, "Symmetry and Symmetry Breaking", The Stanford Encyclopedia of Philosophy (Winter 2017 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2017/entries/symmetry-breaking/.