In the compressor characteristic, what does this value represent?
In German it is called "Ähnlichkeitsparameter", a comparison parameter for the compressor state. This one in particular is a non-dimensional mass flow comparison parameter. To be more correct your equation is actually missing the gas constant "R" under the square root (together with temperature) to make it non-dimensional. This non-dimensional mass flow factor makes it possible to compare different scenarios like high altitude and high speed at low temperatures as well as low speed, sea level conditions. Just like we do with the lift coefficient on wings.

It's not a material property of any particular engine or compressor, just an operating point with a dimensional mass flow m (kg/s), T3 being the total pressure (K) at station 3, P3 being the total pressure at station 3 (Pa) at that time. Station 3 is entry into combustion chamber.

This factor usually comes up in compressor maps (in German "Verdichterkennfeld"), where you plot the compression ratio (pi) on the y-axis over this mass flow parameter on the x-axis. These maps can tell you how much the compressor can compress the air without stalling.

https://en.wikipedia.org/wiki/Compressor_map