Isn't the Pi Pico RP2040 just another microcontroller and lacks the drivers equivalent to those created for the FT232H, or am I missing something?
Yes. Absolutely. Now you can always implement your USB device on a RP2040 as a USB CDC class. Recent OSs will by default attach a virtual COM port to it without requiring a third-party driver. Older OSs (such as Win 7) will require a driver, even for a CDC class. But a virtual COM port interface may not be what you want from an application POV. You can alternatively use a HID class, will not require any driver even on older OSs, but accessing custom HID devices from applications is usually a mess, be prepared to invest a really significant amount of time in all this.
Also, it's not just the driver, you will need to define your own protocol for controlling GPIOs and UART/SPI/I2C... while FTDI also provides ready-to-use libraries. Definitely a lot more work. But also a lot more flexible.
Re-reading your initial post and my answer, I think I may have misunderstood a bit what you wanted to achieve.
I assumed you wanted to replace a RPi with a FT232H, but that's not it, is it? You want to replace a RPi with a computer and a FT232H. Obviously that will not really be a replacement, as a computer, even a laptop, takes up a lot more space and draws a lot more power, but otherwise, why not.
In that case, the only real difference will be again that the FT232H has only one channel and a very limited number of GPIOs you can control, compared to what you have with the RPi. And also you can't use say SPI and UART interfaces at the same time, it's either one or the other.
You can consider the FT2232H which is relatively similar, but with 2 channels. Or go to the FT4232H, which has 4 channels and may get you something closer to what you have with the RPi in terms of number of GPIOs and interfaces.