This question is as much about philosophy as mathematics, however it requires math background so I will ask it here.
I have seen several examples of a mathematical object from analysis or topology being turned into an algebraic concept.
Here are some examples :
- The notion of derivative is first defined using infinitesimal variations. It turns out that by defining an abstract notion of derivation, we can study polynomials derivatives over fields where that notion of "infinitesimal" isn't defined and they still give very useful information (e.g. multiplicity).
- In Algebraic Geometry, we find out that algebraic curves over $\Bbb{C}$ are Riemann surfaces and we can define their topological genus. It turns out that this genus can be defined purely as an algebraic concept.
It raises questions that may seem purely philosophical, but that perhaps have mathematical origins and formulations, and I haven't found any discussion on that anywhere :
- What is the "true nature" of those objects ? Is the algebraic definition a consequence of the analytic / topological definition, or is it the other way ?
- What is the deep reason why those concepts end up making sense in a pure algebraic sense ?
- Would it be possible to give explicit requirements about a mathematical object so that we know that its "true nature" is algebraic ?
The answer to your title question is a simple "no". Algebra has no primacy over topology, or analysis, or other branches of mathematics. So your final question is based on a false premise.
All you are observing with your examples is that algebra can be used to give information about topological or analytic objects. Similarly, topology can be used to give information about analytic or algebraic objects: for example in algebraic geometry one applies topology to study the solutions of equations with complex coefficients (analysis) or of equations with $p$-adic coefficients (algebra and number theory). And similarly, analysis can be used to give information about algebraic or topological objects. And similar interactions occur amongst still more diverse branches of mathematics.
You might be interested in category theory as a tool for applying one branch of mathematics to study another one; perhaps that answers your "deep reason" question.