Use profiles, as @igorlogius said.

If you want different shortcuts, use the following command in your .desktop file (assuming you use Linux):
firefox -no-remote -P

for example, assuming you named your music profile music, you put the following line in a firefox-music.desktop file you’ll create in ~/.local/share/applications/:

Exec=firefox -no-remote -P music (in addition to other common .desktop file fields)

If your firefox executable is not located in your $PATH (for example if you downloaded it directly from Mozilla), then you need to put the path too. for example:
Exec=~/apps/firefox/firefox -no-remote -P music

Edit:

here’s a full .desktop file example for a profile named “work”:
file path and name: /home/confusedllama/.local/share/applications/firefox-work.desktop

Content:

[Desktop Entry]
Name=Firefox for work
GenericName=Web Browser
Exec=/home/confusedllama/apps/firefox/firefox -no-remote -P work
Icon=/home/confusedllama/apps/firefox/browser/chrome/icons/default/default128.png
Terminal=false
Type=Application
MimeType=text/html;text/xml;application/xhtml+xml;application/vnd.mozilla.xul+xml;text/mml;x-scheme-handler/http;x-scheme-handler/https;
StartupNotify=true
Categories=Network;WebBrowser;
Name[en_US]=Firefox for work
Actions=new-window;new-private-window;

[Desktop Action new-window]
Name=Open a New Window
Exec=/home/confusedllama/apps/firefox/firefox -no-remote -P work -new-window

[Desktop Action new-private-window]
Name=Open a New Private Window
Exec=/home/confusedllama/apps/firefox/firefox -no-remote -P work -private-window

This example will give you “Open a new window” and “Open a private window” options too, when you right click on the shortcut. Also as you can see, you can even set a different icon for each shortcut.