I am trying to my PICAXE 08m2 to my laptop using a 3 ring audio jack to USB connector and a TRRS breakout board. I only found out that these are different from the official AXE027 USB Cable Driver. Is there any way to program the 08m2 using what I already have?
The PICAXE download cable is not just a cable. It has an FTDI USB/serial converter chip built in, unlike your cable which is just wired straight through from one plug to the other.
PICAXE's were originally designed to be programed via an RS232 serial port, with a few extra parts to convert the voltage levels from +-12V to the PICAXE's 0-5V logic levels. But modern PCs don't have RS232 serial ports, so they now make a USB cable to do the job. The FT232RQ USB to serial converter chip is programmed to reproduce the signal polarity on an RS232 cable, which is inverted relative to 'standard' TTL serial.
You can use an official FTDI USB/TTL serial cable, but first you must configure it to produce inverted TXD and RXD signals. Alternatively you can use any 'generic' USB/TTL serial converter, with external logic gates or transistors to invert the signals.
Here's an article that describes how to do it:-