Well, I must admit I didn’t really do anything for multisite. I just tested and fixed minor points. Internally, WP makes it easy for plugins to work properly in isolation when used in multisite. I suppose many plugins behave like this (well I have no experience with multisite, so I can’t assert this).
So what you see here is what happens when a single-WP minded plugin is suddenly used in multisite mode.
I don’t know how to set a favicon network-wide. I just didn’t investigate this.
Will the plugin support network-wide favicon? It should, yes. But my TODO list is quite full and that won’t happen before a few months.
In particular, before this, I want to implement automatic favicon upgrade: next time Apple releases yet-another-icon-size along with its brand new iOS 9/X/whatever, your WP favicon will be upgraded automatically.