Is there a drink that's analogous to coffee but made with cocoa beans?
You certainly could brew a drink from pure cocoa powder. It'll be pretty bitter, though; if you don't like unsweetened chocolate you might not like it much, though. So you'll most likely end up adding dairy and sugar, and ending up with hot cocoa.

It won't ever be exactly the same process though. In both cases there's a lot of processing of the beans, and the details differ. But most importantly, while with coffee you can buy and use whole beans, you wouldn't do that with cacao. Cacao beans contain a large amount of fat, so if you want cocoa powder, part of the processing is separating out cocoa solids (which can become cocoa powder) and cocoa butter. If you don't separate them, you end up with a thick paste called chocolate liquor. So you're never going to see whole roasted cacao beans.

But if you're willing to set aside all those differences, I suppose hot cocoa is analogous to coffee, if that means anything.