Play music without a compile

Regards to all! I wonder if the following is feasible: I want music to be played from the SD card via Arduino and certain modules, but I want that when the user removes the SD card and changes the song he wants and returns it to play the song without the need to compile. Is that possible? Again, I have some idea how to program it, but I don't know yet how Arduino works in that regard. If the Arduino can't, which microcontroller can do that?

This highly depends on your program.
If your program is able to - at least - to scan the SD-card for playable files this can work.
More usercomfort will be if you add a display and write code that shows the playable files on the SD_card and the user can choose with some buttons what she/he wants to play.

The more detailed information you give about your setup the more detailed the answers can be

best regards Stefan

so presumably the play-while-replacing-sd-card song is either loaded from the SD if present at startup or hardcoded, and that song is played until the sd card is inserted and a song selected.

If you use something like a DF mini player, your arduino can detect when an SD card is inserted/removed and react accordingly. It can also simply play the first song it sees on the SD card, if a card is present. No re-compile necessary.