While GenServer's store state they are, in my mind, more akin to microservices than Java/C++ Objects. They're like microservices, but without needing a separate service bus, global naming system (pg2), etc. Now I do wish Dialyzer/Dialyxer also had better support for checking GenServer handler's and messages, especially intra-process. You have a point the syntax their could be spruced up some perhaps. The process / GenServer paradigm kind of remind me of Smalltalk in a way, where you're passing messages that object may or may not want to respond to. That's not really possible with C++/Java objects.
GenServers do have their own state, but they are used for way more than just storing state. It's probably best to literally think of it as a server in your running application, analogous to a server running on a network.
If storing state is all you're trying to do you might want to look at Agent or possibly ETS.
Further, GenServers come from Erlang, not Elixir. They're intended to model all sorts of runtime properties of code.
What I like of Elixir is not functional programming, it's the extensive use of pattern matching. What I like less is the convoluted syntax of GenServer with all those handle_cast/calls that obfuscate the real code. As a GenServer is usually a way to store a state, they should have had the courage of calling them objects and give them an object like syntax. After all they initially sold Elixir to developers coming from OO languages. If they had given it a Java like syntax maybe it would be 10 or 100 times as big by now.