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There's an add-on for Firefox which sets different App IDs for different profiles.
There's no registry key for this. It's a Windows bug, and I'm not sure a solution exists.
In general, it happens because Windows replaces a hanged window with another, pseudo Window, with a "Ghost" class. This replacement is synchronized with the taskbar. But when the window gets responsive again, and replaces the "Ghost" window, the replacement isn't always synchronized, and sometimes the taskbar thinks it's a new window.
Looks like it's not the tweaker's fault after all.
I've never encountered such an issue, and I can't think of a reason for the tweaker to screw up thumbnails, especially with the default options set.
Can you reproduce it, or is it random?
You can turn off the tweaked and see if the issue shows up.

Why do you think it's a 7TT issue, and not something else?

You mean this, right?

Not much, it's more about making Firefox more responsive, by e.g. removing unnecessary addons, keeping less tabs open, etc.

Not a bug of the tweaker. See comments.

I'm familiar with this behavior.

A taskbar button moves to the end of the taskbar when it's window hangs for a coupe of seconds or more. It happens mainly with Firefox as it's interface is not very responsive at times.

I'm pretty sure it happens without the tweaker as well, so I'm marking it as not-a-bug.

Right click where?
What is "right-click-button" on keyboard?