Attiny88 as SPI slave
You do not appear to be configuring the slave as an SPI slave. Merely changing the pin modes won't achieve this. Example code at SPI - Serial Peripheral Interface - for Arduino - however that does not specifically mention the Attiny88.

In my example in reply #1 on that page I have this code:

  // turn on SPI in slave mode
  SPCR |= bit (SPE);

  // have to send on master in, *slave out*
  pinMode(MISO, OUTPUT);


This sets SPCR bit MSTR to zero (by not mentioning it). If you call SPI.begin() it will be activated in master mode.

It looks like the Attiny88 has the same register.

So in my opinion your code merely has two SPI masters.

Also, you seem to be doing 3 transfers in the master, but only 2 transfers in the slave.