FT2232H on custom PCB not recognized by PC
It's quite possible that the core voltage cannot handle 3.3V and the chip was fried, so replace the chip (although this is just a hunch, the datasheet is mum on the subject (which is a big problem for not having absolute max ratings listed))
Vcore needs a power filter cap.

+1.8V Output. Integrated voltage regulator output. Connect to VCORE with 3.3uF filter capacitor

Measure on the FTDI side of FB2 and FB3 and make sure the voltage is close to 3.3V (like within 0.1V at minimum) the DC impedance should be in the 10mΩ range (otherwise current will cause a substantial voltage drop across FB2 and FB3. It's a little concerning that they say 600Ω, but that is probably in the MHz range.
Make sure you have decent voltage regulation on the 3.3V bus (your voltage regulator is not shown).
Check the clock and make sure it has good high-low low-high transitions with little ringing.
Edit:
It appears from the added scope screen captures that the max voltage on the OSCI pin is exceeded by 0.7V when it overshoots to 4.31V. This could be causing problems with the clock circuitry which would incapacitate the chip.
[![enter image description here][1]][1]
Before you dig too much into it, redo the scope measurement and measure the clock at the oscillator, make the ground probe as short as possible to minimize inductance that could throw the measurement off, and place the ground probe at the oscillator.
[1]: https://i.stack.imgur.com/BVdBr.png