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

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.

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.

You want to use different profiles.

about: profiles in bar. New profile (for this time only). Launch profile.

Shortcut wise I'm sure it's possible, but I'm not sure

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.

This doesn't work with shortcuts (that I'm aware of) but there's a plugin called "Facebook Container" that sets up virtual browsers within Firefox. Session data, cookies, etc is independant between containers. And it's not just for Facebook, you can setup as many containers as you like (within reason, best to keep the number low) and when you open a link (right click) you can choose which container to open the link in. It's good for isolating web sites (e.g. so Facebook can't see what other websites you're logged into) but also allows you to log into the same website with multiple accounts. For example a personal GMail account and a Work GMail account. Check it out here: https://addons.mozilla.org/en-GB/firefox/addon/facebook-container/

I didn't find a 'profiles' feature like chrome has, but you could install the Firefox beta track, and it'll work independently from your release version

https://docs.telemetry.mozilla.org/concepts/profile/profile_creation.html