I'm afraid such functionality is vendor-specific. I cannot help with other databases, but in Oracle you might probably query the V$SESSION performance view (and count number of rows), although you need quite powerful privileges to be able to do this. There might even be a better dynamic view which would contain the aggregated number of sessions, you might want to look in more detail yourself.
Be careful when interpreting the number you get this way. Some users can have several active connections of their own, and conversely one connection can be shared by many users (if it is a connection managed by connection pool on an application server, for example).
subject: Qry for Number Of User Connected to SQL/Oracle Database.