At about the same time as Floquet, Hill appears to have formulated similar ideas. However, he did not publish them, and the credit of publishing a publicly scrutinizable exposure fairly belongs to Floquet.
Note that there is much more to Floquet theory than what is discussed here. If you have done a course on differential equations, you can see why, since the simplest case of periodic coefficients is constant coefficients. Constant coefficient equations may have exponential solutions that do not have purely imaginary arguments, and they may include algebraic factors if the set of exponentials is not complete. The same happens to the variable coefficient case, with additional periodic factors thrown in. But these additional solutions are not relevant to the discussed periodic crystals. They can be relevant to describing simple crystal boundaries, though.