Have you selected what MEMS microphone chip to use? Analogue or digital? How many bits per sample? What sample rate? If digital what interface?
If the answer are not crazy then I don't know why you'd consider anything other than using a $0.10 32 bit 24-48 MHz microcontroller such as the Puya PY32f002 or WCH CH32V003 rather than some lash-up trying to build a bus using shift registers and logic. The cheapest 8 pin SOIC package might do what you want, or the only slightly more expensive 16 or 20 pin ones certainly will.
Preferably the WCH as a lot more people have experience with them now, there are lots of examples on the internet etc. They run off a wide voltage range (3.3V - 5V, plus another 0.5V either side), don't need a crystal or any other external components (just a bypass cap between GND and Vcc can be useful).
If the bit rate isn't crazy you could use an I2C bus. It's just two wires, one is clock, which is provided by a master which polls up to 127 slave devices. 400 kHz is a standard speed, but there are also 1 MHz and 3 MHz.
With a microcontroller you could collect samples (up to the 2 KB RAM size on the CH32V003) and then return them all when polled, so the polling rate can easily be 100 or 1000 times lower than the sample rate if you want.
Back in the 1980s we used low cost Apple "LocalTalk" networking between PCs (mostly Macs). This used two twisted pairs to network up to 300m at 230.4 kbps. A 3rd party vendor, Farallon, improved on this using a single pair with I think more voltage and you could network over 1800 ft with straight cable or 4500 ft with a single telephone wire pair. There was also a 1 Mbps upgrade. Each computer used a Zilog 8530 SCC (UART). The original LocalTalk and PhoneNet boxes were transformer-coupled, but here is a plan for making compatible but cheaper connections using capacitor coupling instead:
https://www.funet.fi/pub/mac/doc/phone-net-connectors.txt The packet protocols are well documented, and I imagine there are software librareis available.
But on the scale you want, I2C should be fine. I think the I2C spec says 1m maximum, but 2-3m can work.
Note that MEMs microphones all seem to have I2S interfaces, which I think can have multiple devices (channels) on a daisy chain, but is not as flexible as I2C, which is why you need a microcontroller to convert. If microphone chips had I2C then you wouldn't need a per-microphone microcontroller at all, just one for your whole array.