Hi, Marcin,
sorry for the delay, I just come back from a long Chinese festival.
I do not know if the synchronization is needed, it is dependent on your application.
In asynchronous mode, the SAI_RX and SAI_TX are independent module, for SAI_RX, the ADC is master, bit clock(SAI_RX_BCLK) and frame clock(SAI_RX_SYNC) are from ADC module. For SAI_TX module, it can be master, in other words, bit clock(SAI_TX_BCLK) and frame clock(SAI_TX_SYNC) are from SAI_TX module, which can drive the PWR_AMP.
For synchronous mode, the SAI_TX and SAI_RX share the same bit clock and frame clock, for example, you can connect the ADC bit/frame clock to the bit/frame clock of both SAI_TX and SAI_RX, it is synchronous mode, although you can set asynchronous mode for both SAI_RX and SAI_TX. As alternative, you can set the SAI_RX in slave/asynchronous mode, while set the SAI_TX is synchronous mode by setting SYNC bits in I2Sx_TCR2 as 01 in binary, the SAI_TX bit/frame clock are connected to the SAI_RX bit/frame clock internally, SAI_TX bit/frame clcok external pads are not needed.
Hope it can help you
BR
Xiangjun rong