ASP.NET and Gdk are completely unrelated. The former deals with HTTP, the latter deals with screens and native toolkits (X11, wayland, etc.).
The restriction that you saw in the Mono.Cairo homepage is the same as with any Gtk app: don't touch the UI in the incorrect thread. A handy tool to find out this wrong kind of manipulations at runtime in debug mode is gui-thread-check (which is now bundled by default inside Gtk-Sharp3).