Real DOS will preserve the date/time stamp, but DOSBox's lack of support for it isn't really a bug, just something that isn't implemented. There may even be some rationale for it being a good thing that the stamps are changed.
ykhwong's build of DOSBox has support for altering file date/time stamps through INT 21/57, which is useful when using a program, but not when using DOSBox's built-in copy command. So, if you use an external copy program such as XCOPY or the 4DOS command shell, the INT 21/57 support allows the stamps to be preserved when copying. The same also applies to archive programs (PKZIP, LHARC, etc.) which preserve the stamps of files extracted from archives. I'm sure it works on the Windows build, but not sure about the Linux build. http://ykhwong.x-y.net/cvs/frame.html