An executable problem like exe does not work on Linux (without wine). When compiling source code compiler produce object code which is specific to a particular cpu architect
It's due to the difference of how the program is loaded into memory and given resources to run. Even the simplest programs need to have code space, data space and the ability to acquire runtime memory and do I/O. The infrastructure to do these low-level tasks is completely different between the platforms, unless you have some kind of adaptation layer, like WINE or Cygwin.
Assuming, however, that you could just inject arbitrary assembled CPU instructions into the code segment of a running process and get that code to execute, then, yes, the same code would run on either platform. However, it would be quite restricted, and doing complex things like even jumps to external modules would fail, due to how these things are done differently on different platforms.