Is it possible to create a floating context menu item with the exact same font and size as in the native macOS context menu?
Probably yes, would just need to figure out which font and size 
The floating menus support all installed fonts
That’s something I need to figure out then
Another question. Is it possible to make a modifier-based context menu behave like a native one, hiding when a menu item is clicked or when clicking outside?
Currently, the menu appears when I press a modifier key, but I want it to stay visible even after the modifier key is released.
I think it's currently not possible with the standard "Modifier based visibility", but it would be a good addition.
For now you could instead configure your modifier keys in a Key Sequence and use the "Show Floating Menu" action with that.
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Thanks. Good to know it’s not currently possible. I may have spent some time looking for that setting 