I've been messing around with my layout again recently and I am wondering if there is a way to display just a simple text label showing the currently active filter (or static list). Maybe I am overlooking something it but I don't see any way to do it.

Currently the only way is the with_text=1 option of Filter : Filter(with_text=1) this will show the filter name or "static list"/"unnamed filter" next to the filter icon. I guess it could be improved by showing the name of the static list if it is has one, instead of "static list".

Would it be possible to change the font? Or maybe get rid of or change the Icon?

yes, to change the font you can use the markup option, which use the pango markup, %s will be replaced by the text :
markup='<span font="Arial 18">%s</span>'

for the icons, the widget has 3 modes : library filter list
you can change the icon of one or more of these modes with this option :
stock="library:0 filter:gmb-filter-ok list:gtk-edit"
you can use 0 to disable an icon, you can add custom icons in ~/.config/gmusicbrowser/icons/

example with all icons disabled (currently you have to disable each of them individually) :
Filter(with_text=1,markup=<b>%s</b>,stock="library:0 filter:0 list:0")