Yeah search for "firefox profiles", setting everything up how you described it varies depending on your system but it’s definitely possible

I actually run Firefox for all my normal stuff and have a copy of Firefox Developer Edition I use for streaming music and videos so I can keep my "work" and "media" totally separate.

You can have two separate profiles, and two different task bar shortcuts that open up the different profiles. They'd essentially store different user data like history and bookmarks and previously opened tabs, and you could have a music profile and work profile. I did this for work at one point in my life. One helpful thing was making sure they each had different theming to be visually distinct so I could keep track of where to open things.

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.