My Raspberry Zero 2W does not recognize the OLED display

I didn’t use solder or any fasteners, I just held it in place …

You must solder the header pins to the Pi, to ensure a good contact.

The breadboard will be fine and make good contact as it has “clips” inside that are designed to make good contact without needing solder.

However, the Pi only has through-holes, and the copper “pads” around which are not designed to make solderless contact to the pins, they are designed as something for the solder can stick to.

After examining the photo(s) more closely, you also appear to have two single row header pins. Whilst you might be able to get away with just soldering one row of the GPIO pins, it is more usual (i.e. normal) to use a 2×40 male header pin block, rather than two 1×40 headers.

Examine photos of Pi Zero WH units to ensure that you solder the header pins on in the correct orientation. I’ve seen many knackered Pi units for sale, with the headers soldered on the wrong way around, or with female headers, which renders the Pi incompatible with most pHats.

Finally, if you have never soldered before, do not make the header pins on the Pi your first attempt. You will (most likely) make a mess of it (and short out some pins). Practice on an old piece of worthless PCB first, rather than practicing on a ~£20 PCB. Watch a few YouTube tutorials, and remember to check for short circuits between adjacent pins (using a DMM as well as visually).