audio filter designing
"Does audio signal [processing] specifically have any different filtering method?"


well, there are IIR filters that can be implemented in various forms (like Direct I, Direct II, the transposed Direct I or II, Lattice, "State-Variable", Gold-Rader...), there are FIR filters and FIR filters can be implemented the various ways (transversal or the simple summation, Lattice, and "Fast convolution" using the FFT).

any of these filtering methods used on other signal classes can also be used for audio.