What is the purpose of STM32's noise-wave generation function?
This noise generator uses the Linear shift register method to create a pseudo random, maximal length sequence for a 12 bit range. The transition frequency is controlled by the clock selection which determines the upper bandwidth.

The bandwidth is white noise generally flat up to about 1/10 of this clock rate with nulls at 1/2 input clock rate and harmonics of this.

It is useful for performing spectral analysis of any channel within these boundaries by applying the input for spectral analysis.

In CPU Motherboards, random jitter is used on bus clocks to reduce the spectral density in some cases to improve the mrgin of spectral emission test results for FCC/IEC and is often a BIOS option.

In audio systems , white noise is LP filtered with 3dB/ octave to make pink noise used to equalize a room from dominant resonances with suitable mics.  

In RF systems it can be used instead of swept frequency on the modulation to determine channel filter amplitude. 

It can be added to a data channel to accelerate communication errors and thus one can measure the noise margin to error as error margin quick test. This noise margin factor helps isolate other non-random issues of BER problems in the hands of a good analyst such as discrimination error, asymmetry error, pattern error from ISI.

It can be used to measure noise figure of a channel within this range using a bandpass filter, if one is familiar.

Random noise is pseudo in that after the register is reset, the sequence is predetermined so that it can be used for digital data pattern testing on transmit and receive using the data rate as a  clock, a system can be designed to perform BER pattern tests using this algorthm. .  The pattern must begin with some setup that resets the register at the same time at both ends ,  after some command and frame sync is performed.

These are just my experiences of using this LSNG or  PRNG , not on this chip but using the same design.  Other users may have more examples.