When you earn a new privilege it shows you what you earned but not WHERE you earned it (which site), so it then forces you to have to figure that out. It the message actually just said something like "you have earned the privilege to upvote in site" it would be more useful.
There's an icon next to each notification that tells you which site it goes with.
For some sites, the text can be difficult to read or if the site has an icon, it may be difficult to know if you're not familiar with the icons but the information is there already.
Here's a screenshot of the behavior on the iOS app:
- "Lit" is the literature site
- "CB" is our Community Building site
- The empty speech bubble icon between the two red circles (with the +30 next to it) is this site, Meta Stack Exchange.
There's limited space in this pane but I think that using the site icons is a good method for conveying this info, though I admit it takes getting to know the network and recognize the symbols a bit. To confirm the site, you can tap on the achievement and it will take you to the privileges page on the site you earned it on.