I don't believe it's possible to change the size of the icon, unfortunately. However, make sure that the icon your using is a .ico file not some other format. .ico files show up bigger and clearer than other formats.
As for the black icon problem, the color of the taskbar is determined by your Windows color settings. There isn't much you can do other than to make an icon that works for both dark and light colors. Although you could get rid of the default taskbar and make your own taskbar in tkinter with whatever colors you would like, if that's something you would like to undertake.
I have two questions:

Is it possible to increase somehow size of the icon in tkinter? Because the icon choosen by me is not clear visible when I use root.iconbitmap(..).
I also have another icon, but it's black and is not visible on black taskbar. What can I do?
Tkinter - how to change icon size or make black icon visible