This would require dissambling thunderbird, i believe, as the taskbar is by Windows, using the icon provide by Thunderbird internally. There are many websites documenting how to do that, and they work to the extent of being a placeholder for thunderbird, but when clicked the default thunderbird icon appears, making the placeholder useless. If your concern is only to have an icon to start thunderbird that is unique, you don't mind seeing a second icon, then these steps will do that: Here is how:

1. create a desired ico icon file. One way is to locate an old thunderbird one (the classic bird with envelope), probably a jpg or png. then convert from png to ico (several websites do that for free)
2. then, righclick the desktop icon, click properties and the 'change icon' button and set to the new ico file
3. that will change the display on desktop, so you're halfway there
4. now, rightclick and select 'show more options' and select 'pin to taskbar' and the new icon will appear on taskbar.

So far, all is well. But, when you click the custom icon, it does invoke thunderbird, which then plots its default icon on taskbar.

UPDATE: I managed to make this work:

I am assuming windows: I also assume you have created a desired ico file

1. click to the locate of the exe file, thunderbird.exe
2. in that folder is a folder named chrome and in that is a folder named icons and in that folder is a folder named default
3. RENAME the messengerWindow.ico to something else
4. place your icon there, rename to messengerWindow.ico, AND rightclick and make sure that it is UNBLOCKED
5. next, on windows desktop, rightclick the thunderbird icon, select properties and click 'rename'
6. select that icon and click 'apply'
7. done