I have a sketch that detects chip signatures including taking an MD5 sum of the bootloader. Effectively, this gives you a signature of the firmware. Some known signatures are displayed as a known file name. For example:
MD5 sum of bootloader = FB F4 9B 7B 59 73 7F 65 E8 D0 F8 A5 08 12 E7 9F
Bootloader name: optiboot_atmega328
You need another board (eg. a Uno) to run the sketch on.
Not that this will solve your overall problem (unless there is no bootloader at all).
You would also be told the fuse settings, so you could check for something like the "divide clock by 8" fuse being set.