Does the Plugin work with Multisite?

Yes, it works fine in Multisite, you could even activate it Network-wide. However, it does add almost nothing additional to the Network Admin but rather to the Dashboard of each individual sub-site of the Network. Therefore I recommend to use “Toolbar Extras” only in single installs (default) currently.

For Multisite installs I currently recommend my own plugin “Multisite Toolbar Additions” which is like a “sister plugin” of Toolbar Extras (and even was there before it!).

Update I: Toolbar Extras version 1.3.2+ made Multisite support a first-class citizen, including Multisite-aware plugins. That means, if you want to use Toolbar Extras in your Multisite install, then please activate it Network-wide (recommended!); or at least on the Main Site of the Network.

Update II: A special Multisite Add-On for Toolbar Extras is planned but no date is being set yet. This Add-On will bring many improvements to the Network Admin area as well as specific plugin support for Multisite.