The McASP (multi-channel audio serial port) is very flexible, and its parameters can be tweaked to match an ADC/DAC's data format. I think DSP
mode was chosen in the early days because it is the one implemented by the ALSA driver for the audio cape, but I may be wrong. These are highlighted in chapter 22 of the technical reference manual for the AM335x: https://www.ti.com/lit/pdf/spruh73 .
For our internal use and for consultancy projects for commercial applications, we have a C++ API to set those parameters easily, however if you start from our public codebase, you'd have to modify the PRU files instead.
Are the datasheets for these even available online? It would seem by looking at a diagram on their website that they have no on-board crystal or PLL, so you'd need an external one. The Tlv320aic104 codec we use has buit-in PLL, so it can turn the 24MHz (or 12MHz) clock from the McASP into a 44100*32
or 44100*256
bitclock.