All you can test for compatibility with a manual flash is:
Does it fire properly when mounted on the hotshoe when you push the shutter button?
Does it work properly off-camera with the appropriate trigger on your camera hotshoe? (I.e., does it fire in sync, and can you change settings from the trigger that the trigger allows?)
While the YN-560III comes in a single flavor, the RF-602/603/605 triggers that work with it do not--they come in Canon and Nikon flavors, for the single purpose of allowing access to the camera/flash communication protocol's "wake-up" function with the appropriate TTL-capable flashes. While all the other functions work on other brands of camera, that specific function is protocol-specific.
You are correct that manual flashes typically don't have compatibility issues, because only the sync signal is traditionally relayed, but with today's radio triggers there are other issues. For example, the RF-603 (Mk I) triggers only switch to transmit mode if the appropriate non-sync pin signal is communicated. So those triggers cannot be used on non-Canon/Nikon hotshoes, and is why the RF-603II triggers were created with a switch to explicitly set the on-camera unit into transmit mode.
Also understand that the Yongnuo YN-560III is an older model. It's been superseded by the YN-560IV, which in turn is superseded by the YN-660. And that a YN-560-TX is probably the preferred choice of on-camera transmitter over the RF-60x triggers, because it can control the power level remotely on the YN-560III/IV/-660 flashes.
See also: What are the Yongnuo flash naming conventions?