"So, how can we simply reduce physics to a totally algebraic constructs?"
I am not sure it's fair to say that we do this. The data seems to follow these constructs. When the data behaves according to some algebraic construct and its rules then we really luck out. That is an empiricist's view of it.
"Or, how can physical quantities simply emerge from algebraic systems?"
They do not. In my opinion it is somewhat abominable to say algebra begets physical constructs. But that is my opinion.
To understand how we got here you need to understand the history of modern physics. First of all quantum theory was so weird in the 1920's (and still is relative to human experience) that one famous physicist (I think Heisenberg but I'm not sure) said that the math needs to be our intuition. In classical disciplines we can always appeal to our experience and imagine how something would feel or look when we solve a problem. Einstein was quite famous for being able to "feel" the effects of physics, to be able to place himself in a situation and figure out what "should" happen based on physical sense or intuition. In QM no such thing exists in the sphere of human experience! So we go with the math.
Fast forward 20-40 years and we had some understanding of the structure of matter and force fields based on a mathematical symmetry principle involving Lie Groups. Particle physicists became quite adept at this discipline but it was driven by data. The data seemed to fit into a group and we could describe particle decay's and scattering results using the rules imposed by group theory. Theorists started speculating that if such and such a symmetry is there then extending it would lead to several new particles. They predicted and these particles were found. That is not the same as saying that math made physics but I think people became so enamored with the approach it became trendy for theorists to waive a magic math wand and say "look for particles here". It worked! But that's not reality. One example of it not working (yet) is super symmetry. We were expecting to find symmetric partners for all known leptons and force carrying bosons, but we have not yet seen them even though we should have.
A consistent theory of everything may destroy some of the statements made about the connection between the fields of math and their corresponding use in physics. But for now the connection is quite strong.