The most common answer I see is something along the lines of "it's the equivalent of liking a post on twitter". It seems that this is not the case, as the Mastodon devs seem rather adamant that they don't want "likes" in Mastodon. Perhaps it's a method of saving posts? Well, that doesn't make sense either, since there is already the ability to "Bookmark" a post to save it.
It really just seems like a "Favorite" is just a bookmark that tells the poster, and the public that you bookmarked the post. And even if this was the reasoning -- which is baffling enough as it is -- it wouldn't make sense since the whole point of boosting something is to tell the public that you like a post.
It really seems like the "Favorite" button has no actual unique purpose. In my honest opinion, Mastodon should just federate "Likes" like normal, and be done with it.
Sorry, this doesn't make sense. Even the link you post in the OP says it's a like button (literally the first post by Gargron), it just doesn't federate... I'm also not sure if you're aware, but all likes federate to the server that the post is on, so the poster will see all likes they receive, it then just doesn't federate further to everyone else looking at it.
It seems your hang up is that it's called favourite, and making a bunch of assumptions because of this.
Personally I think they should be renamed likes because of situations like this.