Is there a standard on-board diagnostics protocol for FADEC aviation piston engines?
OBD systems were introduced by law in the 80s to reduce / control emissions, so it originated as a method to determine fault with emissions control systems. However, it was refined enough to monitor and control in real time engine and powertrains sytems in further iterations. Keep in mind that the OBDI / OBDII is a protocol that is run on the CANBUS network, which is important to remember that these two are separate entities  (CANBUS can run several different protocols, as Ron Beyer had stated). Other networks such as the LINbus for Audi/VW/BMW run different protocols but still work in a similar fashion.
From my military aircraft maintenance experience (from way back), our systems ran on the 1553 databus, which linked all systems together (nav, comms, AP/FD, etc) and is essentially the CANBUS equivalent. I believe Civil aircraft used the ARINC429 system, however I cannot confirm. For the protocol side, many aircraft (though mainly helicopters) use the Health Usage Monitoring System (HUMS), which provides the OBDII equivilant (more or less) to report back on engine condition, conditional maintenance, operation, etc.
So to make a short story long, there area a few standaridzed networks that aircraft use, however so far there is no standard protocol for all engines. Keep in mind that car manufacturers design and build the system as a whole in-house (chassis, powerplant and drivetrain), so it is not an issue to have all the systems integrate in a relatively simple manner), where airframes are generally designed by a company that will use a powerplant designed and built by a different manufacturer (or perhaps event the same airframe could have different engine variants from different companies), which makes standardization increasingly difficult.