Bootable disks
Bootable floppy disk needs to retain the code in the boot sector. This content cannot be transferred by ordinary file copy utilities. As you already wrote the easiest (and in most cases most reliable) solution is to copy the raw content (all sectors) of the floppy. Besides the boot sector certain system files need to be retained too. The details depends on the operating system which has to be booted from the disk.
Solution with disk images
On Linux and other Unix-like OS you can simply use cat or dd.
Example:
cat /dev/floppy > image.img
cat image.img > /dev/floppy
On Windows it is more complicated. You have to use a third-party tool for reading a floppy to an image and writing the image back to a floppy.
Example with the first set of utilities:
rawread c:\windows\temp\floppy.img A:
rawrite -f c:\windows\temp\floppy.img -d A:
Here are some free command-line utilities: Raw Read/Write Utilities / Utilities to read and write raw diskette images
Here is a shareware with GUI and much more options: WinImage It can be used for 30 -days trial.
The advantages of the "disk image" solution are that it is simple and does not depend on the OS which has to be booted.
The disadvantages are that the disk image could be unnecessarily large, copying could be slower and completely new bootable disk (without an existing disk or image) cannot be created.
Other solutions
Other solution could be to copy all the files and to re-create the boot sector on the destination medium (for example by using command sys on older systems).
Just file copying (without "bootability")
In Windows / MS-DOS there are commands xcopy or robocopy which allow recursive copy of files (including directories and their content).
The File Explorer in Windows performs recursive copy too.
Best Answer
For floppy disks, It isn't free but I would highly recommend Winimage, I recently used this when messing around with PXE disk images and it works very well for this sort of thing.