There is no objects in elixir.
Just because there are no objects in the sense of an Object Oriented (OO) language like C#, functional language still have constructs for collections of data. In Elixir's case, the big three are records, structs and maps. When you would use each is out of the scope of this answer.
How I would receive the data in elixir?
That depends entirely on the semantics of the application and how you are actually using the data in each application.
As an example, if application X needs to ask application Y for a certain subset of its information (Maybe a list of users between the age of 20-30), you may consider setting up some kind of web accessible endpoint in application Y that can filter this information down and return JSON, which application X can then consume and do whatever it needs to with.
An example of a reason you may want to use RabbitMQ, or any other message queue for that matter, would be if you want to do some sort of processing on an event on a different machine than the one the event originated from.
If I would send an object Person from C#.NET to elixir, what I would get in elixir?
C# cannot speak directly to Elixir.