I have this problem too, found this thread after some googling. I'll add my information...

Using latest mIRC (v7.66). It started some months ago, I have never seen it before, now it is always there (but see below). Other than keeping mIRC updated and making tiny script changes I have changed nothing. I have run mIRC on this box for over 10 years. I started out as a Server 2008 but is nowadays a Server 2019 (the last major OS update was long long before this issue appeared).

There are always two extra "window buttons" in the task bar, no more and no less.
They do not appear immediately on program start or when joining servers/channels, but after a few hours they are there. I have never noticed them appearing just when I am doing something on the server, they just turn up when I'm not looking.

I have not found any way to get rid of them, other than quitting mIRC. I would LOVE to know how to remove them. If I right-click on them and choose "close window", the whole application closes. If I can help in any way, I would be happy to do so.