Favimon is described as 'web-based game which lets you battle your favourite websites, building a collection of icons as you attempt to conquer the web'. There are three alternatives to Favimon, not only websites but also apps for Linux, Mac, Windows and Android. The best alternative is Pokemon Showdown, which is both free and Open Source. Other great sites and apps similar to Favimon are Shoddy Battle (Free, Open Source) and Pokemon Laboratory (Free, Open Source).
Pokemon Laboratory is the latest version of the original 4th generation Pokemon simulator, Shoddy Battle. This project is a near 100% rewrite of the original program that strives to be a 100% accurate recreation of Pokemon battle mechanics.