(EXCLAIMER: I've only been able to get Cassettes to load into the game, Vinyl's are not showing but I will update this if I get them to work.

I also have not tested the cassettes with other players online, I can verify they work with objects that accept cassettes, but I am not sure if other players can hear them as well. Again, will update to verify this.)
(UPDATE: I have confirmed that players can hear music you play over a boombox or in-car stereo, and Vinyl's do work, you do need to ensure that the file TCLoadingVinylplayer exists in your mod file, if it doesn't, re-do the steps of adding your music to the mod, ensure that you're adding music to both the cassette and vinyl folders.)

For anyone having issues with getting the mod to work in multiplayer:

If you follow the guide provided by TrueMusic for Multiplayer, the mod sample that is provided leaves out critical files for the mod to function properly.

(If you already have the TrueMusic mod installed, skip ahead a couple paragraphs)

To fix this, it is very simple. First you must subscribe to the TrueMusic mod, and ensure that the mod has installed. To verify that you've installed the mod, navigate to C:\Users\%USERNAME%\Zomboid\Mods, you should see a folder labeled 'TrueMusic'

If you don't see the folder labeled above, if you're adding the mod through Steam, verify that you've Subscribed to the mod, and that Steam has downloaded it (Launching the game should do this automatically.) If you are adding the mod manually, god speed.

( !! Stop skipping here !! )

From here, you can follow the video (https://www.youtube.com/watch?v=b7SUjPzmgfM) until the point it tells you to start uploading the mod to the Steam workshop. This is where you need to manually add some files yourself, while also following the added instructions (you can find these under the Multiplayer instructions on the TrueMusic mod on Steam Workshop) given pertaining to renaming files (NOTE: This only pertains to you if you plan on adding more than 1 TrueMusic Addon, if you're only adding a mod for yourself and friends, ignore the extra naming steps)

Once you've reached the point where the video tells you to start uploading to Steam, you NEED to do the following:

COPY (CTRL + C) the following folders from C:\Users\%USERNAME%\Zomboid\Mods\TrueMusic\media:

\lua\server
\lua\shared
\scripts

and PASTE (CTRL + V) them into your TrueMusic addon folder, it should look something like
C:\Users\%USERNAME%\Zomboid\Workshop\True Music Addon\Contents\mods\truemusic_addon\media

The naming of your Mod may vary, but the idea is still the same. Ensure that you are copying the files from your MOD (The TrueMusic mod) folder to your WORKSHOP (your TrueMusic addon) folder.

To verify that you've done everything correctly, if you navigate to your mod folder (C:\Users\%USERNAME%\Zomboid\Workshop\True Music Addon\Contents\mods\truemusic_addon\media), open up the scripts folder, and there should be items there. Also check the lua folder as well, both the server and shared folder should have contents in them.

Once you've verified that your mod contains all of the necessary files, continue with the upload process and add the mod to your server. (If you don't see any files, go back to the steps above and verify that you're copying the files from the TrueMusic folder in your Mods folder to the TrueMusic addon folder you created in your \Zomboid\Workshop folder)

I have tested this on my own server and it does indeed work. I cannot guarantee that this will work for all of you, your mileage can and will vary but I do hope that this helps some of you as this took me a little bit to figure out.

Cheers!