It won't show up as /dev/usbtiny or something similar.
On my system it is visible as "/dev/bus/usb/002/044" and the udev rule is this:
ATTRS{idVendor}=="1781", ATTRS{idProduct}=="0c9f", MODE="0666" # USBtiny
Without anything connected to the usbtiny the red led flashes briefly when running avrdude. Also the well known error message
avrdude: initialization failed, rc=-1
Double check connections and try again, or use -F to override
this check.
is displayed, which looks pretty much like what you get. Maybe you've just got an error in the wiring or a missing crystal / wrong fuse settings.
Edit:
doesn't your target board come with a mega48 mcu? Therefore '-p m328' can't work.