From the Readme file...
USBasp is built with V-USB driver by OBJECTIVE DEVELOPMENT GmbH. See
"firmware/usbdrv/" for further information.
I'm confused a bit...
Yes this USBASP programmer ($1.19 from AliExpress) can be programmed w/ Arduino but with 1K SRAM, 8K flash it is 2x the SRAM of the tiny85 but with more I/O. Unfortunately, getting to that I/O means soldering wires to the breakout board or directly to the uC.
Since V-USB is used to connect to the USB PC driver, would it not make more sense to use an Atmega328 w/ 16MHz crystal and a V-USB enabled bootloader. Oh, my... that sounds just like Adafruit's Pro-Trinket!
Here is how I cloned it: Tchotchke (Pro Trinket clone) USB Programming Stick
Ray