How to find out ALL the users that can connect to SSAS? I have an user called mycompany\aives that is not listed in the list of administrators in the SSAS, not belong to any of the AD (active directory) groups listed there, 

however,

he can connect to the cubes in this server (called SASBI01).

if you see , however, the second picture, mycompany\aives is listed as one of the administrators on the box (SASBI01)

Where does it say that the administrators on the box can (or not) connect to the cubes?

How can I find out (preferably via script), a list of users that can connect to the cubes and their permissions?
Check the server property BuiltinAdminsAreServerAdmins. Till SQL Server 2014 it defaults to true. Not sure if it will be  changed in 2016 (hopefully).

Link: https://msdn.microsoft.com/en-us/library/ms174561(v=sql.120).aspx